Sharp lower bound for the total number of matchings of graphs with given number of cut edges
|
|
- Alban Leonard
- 5 years ago
- Views:
Transcription
1 South Asian Journal of Mathematics 2014, Vol. 4 ( 2 ) : ISSN RESEARCH ARTICLE Sharp lower bound for the total number of matchings of graphs with given number of cut edges Hongzhuan Wang 12, Rongrong Gu 2 1 Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huai an, Jiangsu , P.R. China 2 Faculty of Foreign Languages, Huaiyin Institute of Technology, Huai an, Jiangsu , P.R. China hongzhuanwang@gmail.com Received: March ; Accepted: April *Corresponding author This research was partially supported Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (No. 12KJB110001). Abstract The total number of matchings of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent. In this paper we determine a sharp lower bound for the total number of matchings among the set of graphs with k cut edges for all possible values of k, and characterize the corresponding extremal graphs as well. Key Words MSC 2010 Total number of matchings; Cut edge; 2-Edge-connected graph; Extremal graph 05C90, 05C50 1 Introduction In the chemistry literature total number of matchings is called the Hosoya index of a molecular graph. If we denote by m(g, k) the number of matchings with k edges in G, then its Hosoya index Z(G) can be expressed as Z(G) = n 2 k=0 m(g, k), where n is the number of vertices of G and n 2 is the greatest integer n 2. As a chemical descriptor of molecular structures, the Hosoya index has received much attention in the literature since it was introduced by Hosoya [4]. An important direction is to determine the graphs with maximal or minimal Hosoya indices in a given class of graphs. In [7] Gutman showed that the linear hexagonal chain is the unique chain with minimal Hosoya index among all hexagonal chains. In [11] Zhang showed that the Zig-zag hexagonal chain is the unique chain with maximal Hosoya index among all hexagonal chains. In [12] Zhang and Tian gave another proof of Gutman s and Zhang s results above mentioned. In [13] Zhang and Tian determined the graphs with minimal and second minimal Hosoya indices among catacondensed Citation: Hongzhuan Wang, Rongrong Gu, Sharp lower bound for the total number of matchings of graphs with given number of cut edges, South Asian J Math, 2014, 4(2),
2 H. Wang, et al: Sharp lower bound for the total number of matchings of graphs with given number of cut edges systems. In [8] the path and star have been shown to have the maximal and minimal Hosoya indices, respectively, among all trees on n vertices. Hou [16] characterized the trees having minimal and second minimal Hosoya indices among all trees with a given size of matching. Yu et al. [1] investigated the graphs having minimal Hosoya index among all graphs with given edge independence number and cyclomatic number. In [2] Yu et al. investigated the trees having minimal Hosoya index among all trees with k- pendent vertices. In [14] Li, Li and Zhu determined the n-vertex unicyclic graphs with the minimal, second-, third-, fourth-, fifth- and sixth-minimal Hosoya indices. In [3] Heuberger and Wagner gave a characterization of the trees with given maximum degree which maximize the number of independent subsets, and showed that these trees also minimize the number of independent edge subsets. Zhu and one of the present authors [15] characterized the unique unicyclic graphs of a given diameter with the maximum number of independent sets. In order to state our results, we introduce some notation and terminology. For other undefined notation we may refer to Bondy and Murty [9]. We only consider finite, undirected and simple graphs. For a vertex v of a graph G, we denote N(v) = {u uv E(G)} and N[v] = N G (v) {v}. We denote d(v) be the cardinality of N(v). A pendent vertex is a vertex of degree one of G. If e is an edge in G incident with one pendent vertex, we call e a pendent edge. The graph that arises from G by deleting the vertex u V (G) or the edge uv E(G) will be denoted by G u or G uv. As usual, P n, C n and S n denote a path, a cycle and a star on n vertices, respectively. A cut vertex of a connected graph G is a vertex whose deletion results in a disconnected graph. A cut edge is defined similarly. A connected graph without cut edges is also called 2-edge-connected graph. We denote by K m,n the complete bipartite graph whose partition sets are of size m, n respectivly. If G is a connected graph with k cut edges, then clearly 1 k n 1 and k n 2. Let C n k (1 k ) be the graph obtained from C n k by attaching k pendent edges to one of its vertices. We denote G 0 (n, k) be the graph obtained from K 2,n k 2 by attaching to one of its maximum-degree vertices k pendent edges. The path P t = v 1 v 2 v t is called a pendent path of G if d(v 1 ) 3, d(v 2 ) = = d(v t 1 ) = 2 and d(v t ) = 1 in G. Let F n denote the nth Fibonacci number. Then we have F n + F n+1 = F n+2, with initial conditions F 1 = F 2 = 1. Let tg stand for the disjiont union of t copies of G. To our best knowledge, the total number of matchings of graphs with given cut edges was, so far, not considered. In this paper we investigate the total number of matchings for the set of graphs with k cut edges. In next section, we give some preliminary results. In Section 3, we first present a sharp lower bound on the total number of matchings among the set of the graphs without cut edges and characterize the extremal graphs achieved the bound. Secondly, we characterize the graphs with the smallest value of total number of matchings among the set of graphs with k ( 1) cut edges for different values of k. In Section 4, we give a conclusion of this paper. 2 Some preliminary results In this section, we list some necessary results which are needed in this paper. Lemma 2.1 ([8]). Let G = (V, E) be a graph. Then 108
3 South Asian J. Math. Vol. 4 No. 2 (i) If uv E(G), we have Z(G) = Z(G uv) + Z(G {u, v}); (ii) If v V (G), we have Z(G) = Z(G v) + Z(G {u, v}); u N(v) (iii) If G 1, G 2,, G t, (t 1) are the components of graph G, we have Z(G) = t Z(G j ). For a graph G, according to the definition of Z(G), by Lemma 2.1 (ii), if v is a vertex of G, then Z(G) Z(G v). In particular, when v is a pendent vertex of G and u is the unique vertex adjacent to v, we have Z(G) = Z(G v) + Z(G {u, v}). Corollary 2.2. Let G be a graph with e E(G), then we have Z(G e) < Z(G). Lemma 2.3 ([16]). Let T be a tree on n vertices. Then Z(S n ) Z(T ) Z(P n ), and Z(T ) = n if and only if T = S n and Z(T ) = F n+1 if and only if T = P n. Lemma 2.4 ([5]). If G 1 is a proper subgraph of G 2, then Z(G 2 ) > Z(G 1 ). Lemma 2.5 ([10]). If G is a connected unicyclic graph on n vertices, then Z(G) 2n 2, with equality if and only if G = Sn, 3 where Sn 3 denoted by the graph obtained by adding one edge between two pendent edges of star with n vertices. Lemma 2.6 ([15]). Let X, Y and Z be three connected graphs with disjoint vertex sets and let u, v V (X), v 0 V (Z), u 0 V (Y ). Let G be the graph obtained from X, Y and Z by identifying v with v 0 and u with u 0, respectively. Let H be the graph obtained from X, Y and Z by identifying three vertices v, v 0 and u 0, and let J be the graph obtained from X, Y and Z by identifying three vertices u, v 0 and u 0 (see Fig. 1). Then Z(G) > Z(H) or Z(G) > Z(J), where G, J, H are all connected graphs having at least four vertices. Y Y u X v u v H X Z Z G Y u J Z X v Fig 1: The graphs in Lemma 2.6 From Lemma 2.6, the following two results are obvious, we omit their proofs. Lemma 2.7. Let uv E(G) be a cut edge of a graph G and G uv = G 1 G 2, where u V (G 1 ) and v V (G 2 ). Let G be the graph obtained from G by identifying vertex u with v (the new vertex is labeled as w) and attaching at w a pendent vertex w 0. Then Z(G) > Z(G ). 109
4 H. Wang, et al: Sharp lower bound for the total number of matchings of graphs with given number of cut edges Lemma 2.8. Let G be a connected graph with u, v V (G). Denote by G(s, t) the graph obtained by attaching s 0 pendent vertices to vertex u of G and t 0 pendent vertices to vertex v of G. Then Z(G(s, t)) > Z(G(0, s + t)) or Z(G(s, t)) > Z(G(s + t, 0). 3 Main results In the following subsection, we first consider the 2-edge-connected graphs, namely, graphs without cut edges. 3.1 The smallest values of total number of matchings in 2-edge-connected graphs Let Span(G) denote the spanning tree of a graph G. Let F(n) denote the set of 2-edge-connected graphs on n vertices. If n = 3, then F(3) contains a single element C 3, so we always assume that n 4 in the remainder part of this section. Lemma 3.1. Let G be a connected graph. If G is not a tree, then Z(G) 2n 2 and this equality holds if and only if G = C 3 (1 n 3 ). Proof. Let T be a spanning tree of G and e an edge in E(G\T ). Then G = T + e is a unicycle spanning subgraph of G. Hence, by Lemmas 2.4 and 2.5, we have that Z(G) Z(G ) 2n 2, and equality holds if and only if G = C 3 (1 n 3 ). Lemma 3.2. If t 2 and n j 2 (j = 1, 2,, t), then Z( t S nj ) Z(S t ), and equality holds if n j and only if n j = 2 and t = 2. Proof. We shall proceed by induction on t. First, consider the case of t = 2. In this case, we have Z(S n1+n 2 ) Z(S n1 S n2 ) = (n 1 + n 2 ) n 1 n 2 = 1 (n 1 1)(n 2 1) 0. Now, suppose that t 3 and the statement is true for smaller values of t. Thus, by Lemma 2.1 (iii) and the induction hypothesis, we have Z( t S nj ) Z( t 1 S n j )Z(S nj ) Z(S t 1 )Z(S nt ) Z(S t ), n j n j as desired. The equality holds if and only if n j = 2, t = 2. We first choose G min F(n) such that the Hosoya index of G min is as small as possible. Now we will show some properties of graph G min. when 4 n 7, by direct calculation, G min = Cn. When n = 8 and n = 9, by direct calculation G min = K2,6 and G min = K2,7 respectively. In the following we assume that n 10. Lemma 3.3. G min C n for n
5 South Asian J. Math. Vol. 4 No. 2 Proof. Suppose that G min = Cn for n 10. A simple computation gives that for n 10. Z(G min ) = Z(C n ) > Z(K 2,n 2 ), contradiction to the choice of G min. Lemma 3.4. For any edge e in G min, G min e contains at least one cut edge. Proof. Suppose that G min e has no cut edge. Then by Corollary 2.2, Z(G min e) < Z(G min ), contradicting the choice of G min. Lemma 3.5. Suppose that G C n, for any edge e E(G min ), If G min e contains at least a non-pendent cut edge, then Z(G min ) > n 2 3n + 3. Proof. Let e = uv be an edge in G min such that G min e contains a non-pendent cut edge. We first know that G min {u, v} is a connected graph, otherwise, if G min {u, v} is not connected, then Z(G min {u, v}) 1, the equality holds if all vertices in G min {u, v} are isolated vertices. Therefore, we deduce that G min = K2,n 4 + e, then Z(G min ) > Z(K 2,n 4 ), contradiction to the choice of G min. Next, we consider the following two cases. Case 1. G min {u, v} is a tree. In this case, by Lemma 2.3, Z(G min {u, v}) Z(S n 2 ) = n 2. For G C n, then G min has at least two cycles, then e must be one edge of a cycle. If G min e contains non-pendent cut edges, that is to say, there exist a pendent path of length at least 2 in G min e. When G min {u, v} = S n 2, firstly, all the pendent vertices of S n 2 must be adjacent to u or v, otherwise, there exist cut edge in G min, contradiction to the choice of G min. Secondly, u, v are both adjacent to the pendent vertices of S n 2, otherwise, then is no pendent path of length at least 2 in G min e, contradiction to the assumption. Thirdly, we conclude that one vertex of u and v is adjacent to one pendent vertices of S n 2 and another is adjacent to all the remainder pendent vertices of S n 2, if not, G min e has no cut edges, contradiction to Lemma 3.4, then Z(G min e) Z(G ) = 2n 2 13n + 23, where G is the graph obtained from K 2,n 4 by attaching to one of its maximum degree vertices a pendent path of length 2, (see Fig. 2). By Lemma 2.1, we have Z(G min ) = Z(G min e) + Z(G min {u, v}) 2n 2 13n n 2 > n 2 3n + 3 (n 10). Case 2. G min {u, v} is not a tree. In this case, by Lemma 3.1, Z(G min {u, v}) Z(C 3 (1 n 3 )) = 2n 2, if G min e contains a non-pendent cut edge, similar as argument in Case 1, we have Z(G min e) Z(G ) = 4n 2 38n + 94, where G is the graph obtained by identifying the pendent vertex of P 3, one vertex of C 3 and one maximum degree vertex of K 2,n 6, (see Fig. 2). By Lemma 2.1, Z(G min ) = Z(G min e) + Z(G min {u, v}) 4n 2 38n n 2 > n 2 3n + 3 (n 10). 111
6 H. Wang, et al: Sharp lower bound for the total number of matchings of graphs with given number of cut edges u v G u u G Fig 2: The graphs in Lemma 3.5 This proves Lemma 3.5. By an elementary calculation, we obtain that Z(K 2,n 2 ) = n 2 3n + 3, thus, by Lemma 3.5, all cut edges in G min e must be pendent edges. If not, Z(G min ) > Z(K 2,n 2 ), a contradiction to our choice of G min. Furthermore, we claim that all cut edges in G min e are pendent edges attached to one common vertex of a 2-edge-connected graph. Since G min is a 2-edge-connected graph, G min e has at most two pendent edges. If there is only one pendent edges in G min e, the claims holds immediately. If there are two pendent edges, by contradiction assume that the pendent edges in G min e attached to different vertices of a 2-edge-connected graph, then move the pendent edges to the common vertex, we obtain a new graph G such that G + e is a 2-edge-connected graph, by Lemma 2.1 (i) and Lemma 2.8, we have that Z(G + e) < Z(G min ), then contradiction to the choice of G min. From above argument, the following results holds immediately. Corollary 3.6. For any e E(G min ), all cut edges in G min e are pendent edges and attached to one common vertex of a 2-edge-connected graph. Lemma 3.7. For any edge e in G min, G min e contains only one cut edge. Proof. Suppose to the contrary that G min e contains two cut edges. In what follows we shall show that Z(G min ) > n 2 3n + 3 = Z(K 2,n 2 ), while K 2,n 2 is a 2-edge-connected graph, which contradicting the choice of G min. If G min e contains two cut edges, then G min must be the graph as depicted in G 1 (see Fig. 3). Let e = uv be any edge of G min, we denote by H = G min {u, v} the 2-edge-connected subgraph of G min, we have that N H (u) = N H (v) = x. Note that Z(G min ) = Z(G min uv) + Z(G min {u, v}) = Z(G min e) + Z(H). (1) In Equation (1), if Z(G min e) and Z(H) attain the least at the same time, the inequality Z(G min ) > n 2 3n + 3 holds, then any other cases ensure the inequality holds clearly. Therefore, in the following, we assume that the value of Hosoya index of Z(G min e) and Z(H) are as small as possible. Next, we distinguish two steps to obtain our result. First, we consider the Hosoya index of G min e in Equation (1). In the following, we shall prove that Z(G min e) n 2 5n
7 South Asian J. Math. Vol. 4 No. 2 By Lemma 2.1 (ii) and G 1 (see Fig. 3), we have Z(G min e) = Z(G min e x) + Z(G min e {x, y}) y N(x) = Z(H x) + Z(H {x, y}) + 2Z(H x) y N H (x) = 3Z(H x) + Z(H {x, y}). (2) y N H (x) Since H is 2-edge-connected graph, H x has no isolated vertex. For the above chosen vertex x, we consider the following cases: Case 1. Suppose that H x and H {x, y} is connected. Then Z(H x) Z(Span(H x)) n 3. We know that H {x, y} attains the smallest Hosoya index when H {x, y} = S n 4, and d H (x) = n 4, where y N H (x). By above statement and Equation (2), we have Z(G min e) 3(n 3) + (n 4)(n 4) = n 2 5n + 7. Case 2. Suppose that H x has t (t 2) components, say G 1, G 2,..., G t, and for any vertex y N H (x), H {x, y} has s (s 2) components and has no isolated vertices. Let V (G j = n j for 1 j t. Then, from Lemmas 2.3 and 2.4, it follows that Z(G j ) Z(Span(G j )) Z(S nj ). Again by Lemmas 2.1 (iii), 2.3, 2.4 and 3.2, we have t t Z(H x) = Z( G j ) = Z(G j ) t Z(Span(G j )) t t Z(S nj ) = Z( S nj ) Z(S t ) = Z(S n 3 ) = n 3. n j Similarly, we can prove that Z(H {x, y}) n 4. By above statement and Equation (2), we obtain Z(G min e) 3(n 3) + (n 4)(n 4) = n 2 5n + 7. Case 3. Suppose that H x is connected and for any vertex y N H (x), H {x, y} has s (s 2) components without isolated vertices. By using the same method as above, we can obtain that Z(H x) Z(Span(H x)) n 3, and Z(H {x, y}) n 4. Hence, by Equation (2), we have Z(G min e) 3(n 3) + (n 4)(n 4) = n 2 5n + 7. Case 4. Suppose that there exists a vertex y N H (x) such that Z(H {x, y}) has at least an isolated vertex and let w be an isolated vertex in Z(H {x, y}). Then d H (w) = 2, that is, N H (w) = {x, y}, and 113
8 H. Wang, et al: Sharp lower bound for the total number of matchings of graphs with given number of cut edges so w N H (x). By the definition of Z(G), it is easy to see that the more vertices as w, the smaller Hosoya index of G min e. Then, if all the vertices of H {x, y} are isolated, the Hosoya index of G min e attains minimum. Hence G min e must be the graph as depicted in G 2 (see Fig. 3). So, by Equation (2), we have Z(G min e) 3(n 3) + (n 4)(n 4) + 1 = n 2 5n > n 2 5n + 7, thus the proof of the first step is completed. Secondly, we consider the Hosoya index of H in Equation (1). In the following, we shall prove that Z(H) > 2n 4. In order to prove the inequality, we first distinguish the following assumptions. Assumption 1. For any e = u v E(H), the graph H e is connected and contains at least a cycle. Indeed, if H e is a tree, in view of H is a 2-edge-connected graph, then H = C n 2. Hence, the result follows from a direct calculation for n 10, Z(H) = Z(C n 2 ) > Z(K 2,n 4 ), which contradicting the previous assume of Z(H). Thus Assumption 1 follows. By Assumption 1 and Lemma 3.1, we obtain that Z(H e) 2(n 2) 2, but then H e C 3 (1 n 5 ), contradicting the choice of H. Consequently, Z(H e) > 2(n 2) 2 = 2n 6. Assumption 2. For any e = u v E(H), we have that Z(H {u, v }) 2. Otherwise, if Z(H {u, v }) = 1, then all the vertices of H {u, v } are isolated. Hence H = K 2,n 4 + e. Note that K 2,n 4 + e contains K 2,n 4 as its proper subgraph. So we obtain that Z(H) > Z(K 2,n 4 ), which also contradicting the previous assume of Z(H). By combining Assumptions 1 and 2, we have Z(H) = Z(H e) + Z(H {u, v }) > 2(n 2) = 2n 4, as desired. From the combination of two steps above, Lemma 2.1 (i) and Equation (1), it follows that Z(G min ) = Z(G min e) + Z(H) > n 2 5n n 4 = n 2 3n + 3. But this contradicting the choice of G min. This completes the proof of Lemma 3.7. e u x u x H y v v H v u e v G 1 G 2 G 3 Fig 3: The graphs in Lemma 3.7 and Theorem 3.8 Now we prove the main result of this section. 114
9 South Asian J. Math. Vol. 4 No. 2 Theorem 3.8. If G is any graph in F(n) with n 8, then Z(G) n 2 3n + 3, and equality holds if and only if G = K 2,n 2. Proof. In view of Lemmas 3.4, 3.7, and Corollary 3.6, we obtain that G min e must has exactly one pendent edge. Let e = uv be any edge in G min such that G min e contains only one pendent edge. Then one of u and v must have degree 2 in G min. We may assume that d Gmin (v) = 2. Without loss of generality, suppose vv E(G min ) and let H = G min v, where H is a 2-edge-connected graph. It is evident that u H. In this case, G min is the graph shown in G 3 (see Fig. 3). We proceed by induction on n. First, consider the validity of the above statement for the case of n = 8. By the properties of G min and direct calculation, we have that Z(G min ) > Z(K 2,6 ), the assertion is true. Now let n 9 and suppose that the above statement is true for smaller values of n. By Lemma 2.1 (ii) and G 3 (see Fig. 3), we obtain Z(G min ) = Z(G min v) + Z(G min {u, v}) u N(v) = Z(G min v) + Z(G min {u, v}) + Z(G min {v, v }) = Z(H ) + Z(H u) + Z(H v ). (3) Thus by Equation (3), we distinguish three cases to obtain our result. For H is 2-edge-connected graph, then H u and H v have no isolated vertices. Case 1. Both H u and H v are connected. By Lemmas 2.3 and 2.4, we have Z(H u) Z(Span(H u)) Z(S n 2 ) n 2. Similarly, by Lemmas 2.3 and 2.4, we can show that Z(H v ) n 2. Again by induction hypothesis, we have that Z(H ) (n 1) 2 3(n 1) + 3. In view of above statement and Equation (3), we obtain Z(G min ) (n 1) 2 3(n 1) (n 2) = n 2 3n + 3, equality holds if and only if H u = S n 2, H v = Sn 2, H = K2,n 3. Case 2. Suppose that H u is connected, while H v has r (r 1) components. By Lemmas 3.2, 2.3 and 2.4, we have that Z(H v ) n 2 and, by Lemmas 2.3 and 2.4, we further obtain that Z(H u) Z(Span(H u)) Z(S n 2 ) n 2. Therefore, again by induction hypothesis, above statement and Equation (3), we have that Z(G min ) n 2 3n + 3, and this equality holds if and only if Z(H u) = n 2, Z(H ) = (n 1) 2 3(n 1) + 3, Z(H v ) = n 2, that is, H u = S n 2, H v = S2 S2, H = K2,n
10 H. Wang, et al: Sharp lower bound for the total number of matchings of graphs with given number of cut edges If H v = S 2 S2, then G min must be the graph as depicted in G 4 (see Fig. 4), but then, H K 2,n 3, H u S n 2. Hence, Z(H ) > (n 1) 2 3(n 1) + 3 and Z(H u) > n 2, and thus Z(G min ) > n 2 3n + 3. That is, the equality doesn t holds in Case 2. Case 3. Suppose that H u is not connected, while H v is connected. we can use the same method as above to show that Z(G min ) > n 2 3n + 3. Consequently, by Cases 1-3, the equality holds if and only if H u = S n 2, H v = Sn 2, H = K2,n 3, which is equivalent to G min = K2,n 2. Therefore, the proof of Theorem 3.8 is completed. In Section 3.2, we shall consider the graph with at least one cut edge. 3.2 The smallest values of total number of matchings in graphs with k ( 1) cut edges Let H(n, k) denote the set of connected graphs with n vertices and k cut edges. Also, we use (kp 2 )vh to denote the graph arisen from H by pasting k paths P 2 to the vertex v of H (see (kp 2 )vh in Fig. 4 for instance). u H v v v G 4 G 5 (kp 2 )vh Fig 4: The graphs in Theorem 3.9 Theorem 3.9. Let any G H(n, k) with 1 k n 4 and n 8. Then Z(G) (n 1)+(n 2)(n k 2). Equality holds if and only if G = G 0 (n, k), where G 0 (n, k) is the graph obtained from K 2,n k 2 by attaching to one of its maximum-degree vertices k pendent edges. Proof. Let G min be chosen from H(n, k), such that Z(G) Z(G min ) for any G in H(n, k). Next, we shall prove that G min = G 0 (n, k). Suppose to the contrary assume that G min G 0 (n, k). From Lemma 2.6, Lemma 2.7 and Lemma 2.8, it follows that all cut edges in G min are pendent edges attached to one common vertex of a 2- edge-connected graph H with n k vertices. Moreover, G min has exactly one cut vertex, say v. Thus G min = (kp2 )vh, as depicted in Fig. 4. In the following, we shall prove that Z(G min ) = Z((kP 2 )vh) > Z(G 0 (n, k)) by contradiction. 116
11 South Asian J. Math. Vol. 4 No. 2 If k = 1, then we have Z(G 0 (n, 1)) = Z(K 2,n 3 ) + Z(S n 2 ). (4) Z(P 2 vh) = Z(H) + Z(H v). (5) Since H is a 2-edge-connected graph with n 1 vertices, Z(H) Z(K 2,n 3 ) with equality if and only if H = K 2,n 3 by Theorem 3.8. If H v is connected, then, by Lemmas 2.3 and 2.4, Z(H v) Z(Span(H v)) Z(S n 2 ), with equality if and only if H v = S n 2. By our assumption that G min G 0 (n, k) and Equations (4-5), we have that Z(G min ) = Z(P 2 vh) > Z(G 0 (n, 1), which contradicting the choice of G min. Assume that H v has components Q 1, Q 2,, Q t (t 2). Let n j denote the order of Q j for j = 1, 2,, t. Similar to above, we have that Z(Q j ) Z(Span(Q j )) Z(S nj ). By Lemmas 2.3 and 2.4, we obtain t t Z(H v) Z(S nj ) = Z( S nj ) Z(S n 2 ), with equality if and only if H v = 2S 2, namely, G min is isomorphic to the graph G 5 (see Fig. 4). But then Z(G 5 ) > Z(G 0 (6, 1)), a contradiction. Hence Z(H v) > Z(S n 2 ). By Equations (4-5), we have that Z(G min ) = Z(P 2 vh) > Z(G 0 (n, 1)), again a contradiction. If k 2, then by Lemma 2.1 (ii), we obtain the following recursion relations: Z(G 0 (n, k)) = Z(G 0 (n 1, k 1)) + Z(S n k 1 ). (6) Z(G min ) = Z((kP 2 )vh) = Z(((k 1)P 2 )vh) + Z(H v). (7) Combining Equations (6-7) and the initial condition Z(P 2 vh) > Z(G 0 (n 1, 1)), we have that Z(G min ) = Z((kP 2 )vh) > G 0 (n, k) for all 2 k n 4, since G min G 0 (n, k). So Z(G min ) = Z((kP 2 )vh) > G 0 (n, k) for all 1 k n 4, which contradicting the choice of G min. This contradiction gives G min = G 0 (n, k) for all 1 k n 4. It is not difficult to verify that Z(G 0 (n, k)) = n 1+(n 2)(n k 2). Therefore, the proof of Theorem 3.9 is completed. By a similar discussion as in the proof of Theorem 3.9, we may also show the following result is true. We omit the procedure here. Theorem For any G H(n, k) with 1 k n 4 and 4 n 7. Then Z(G) (k + 1)F n k + 2F n k 1. Equality holds if and only if G = C n k (1 k ). 4 Conclusion In this paper we characterized the graphs with the smallest values of total number of matchings among the set of graphs with k cut edges for different values of k. It is surprised to see that the graph on n vertices graphs with given cut edges which attains the smallest values of total number of matchings is not unique. When 4 n 7 there exist one case and when n 8, there is another case. Let any G H(n, k) with 0 k n 1, n 4 and k n 2. Then the following statements hold. If k = n 1, then Z(G) n equality holds if and only if G = S n ; 117
12 H. Wang, et al: Sharp lower bound for the total number of matchings of graphs with given number of cut edges If k = n 3, then Z(G) 2n 2 equality holds if and only if G = C 3 (1 n 3 ); If k = 0 and 4 n 7, then Z(G) F n+1 + 2F n equality holds if and only if G = C n ; If k = 0 and n 8, then Z(G) n 2 3n + 3 equality holds if and only if G = K 2,n 2. If 1 k n 4 and 4 n 7, then Z(G) (k + 1)F n k + 2F n k 1 equality holds if and only if G = C n k (1 k ); If 1 k n 4 and n 8, then Z(G) (n 1) + (n 2)(n k 2) equality holds if and only if G = G 0 (n, k). On the other hand, it is natural to consider the following problem which may be much more difficulty. Problem 4.1 How can we determine a sharp upper bound on the total number of matchings of graphs with given cut edges. References 1 A. M. Yu, F. Tian, A kind of graphs with minimal Hosoya indices and maximal Merrield-Simmons indices, MATCH Commun. Math. Comput. Chem. 55(1) (2006) A. M. Yu, X. Z. Lv, The Merrield-Simmons indices and Hosoya indices of trees with k pendent vertices, J. Math. Chem. 41 (2007) C. Heuberger, S. G. Wagner, Maximizing the number of independent subsets over trees with bounded degree, J. Graph Theory 58(1) (2008) H. Hosoya, Topological index, Bull. Chem. Soc. Jpn. 44 (1971) H. B. Hua, Hosoya index of unicyclic graphs with prescribed pendent vertices, J. Math. Chem. 43(2) (2008) H. Liu, M. Liu, A unified approach to extremal cati for different indices, MATCH Commun. Math. Comput. Chem. 58 (2007) I. Gutman, Extremal hexagonal chains, J. Math. Chem. 12 (1993) I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, Macmillan, New York, J. P. Ou, On extremal unicyclic molecular graphs with prescribed girth and minimual Hosoya index, J. Math. Chem. 42(3) (2007) L. Z. Zhang, The proof of Gutman s conjectures concerning extremal hexagonal chains, J. Sys. Sci. Math. Sci. 18 (1998) L. Z. Zhang, F. Tian, Extremal hexagonal chains concerning largest eigenvalue, Sci. China A 44 (2001) L. Z. Zhang, F.Tian, Extremal catacondensed benzenoids, J. Math. Chem. 34 (2003) S. Li., X. Li, Z. Zhu, On minimal energy and Hosoya index of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 61 (2009) S. Lin, C. Lin, Trees and forests with large and small independent indices, Chinese J. Math. 23 (1995) Y. Hou, On acyclic systems with minimal Hosoya index, Discr. Appl. Math. 119 (2002)
ON THE SUM OF THE SQUARES OF ALL DISTANCES IN SOME GRAPHS
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 38 017 (137 144) 137 ON THE SUM OF THE SQUARES OF ALL DISTANCES IN SOME GRAPHS Xianya Geng Zhixiang Yin Xianwen Fang Department of Mathematics and Physics
More informationA NOTE ON THE NUMBER OF DOMINATING SETS OF A GRAPH
A NOTE ON THE NUMBER OF DOMINATING SETS OF A GRAPH STEPHAN WAGNER Abstract. In a recent article by Bród and Skupień, sharp upper and lower bounds for the number of dominating sets in a tree were determined.
More informationOn the Relationships between Zero Forcing Numbers and Certain Graph Coverings
On the Relationships between Zero Forcing Numbers and Certain Graph Coverings Fatemeh Alinaghipour Taklimi, Shaun Fallat 1,, Karen Meagher 2 Department of Mathematics and Statistics, University of Regina,
More informationOn vertex-coloring edge-weighting of graphs
Front. Math. China DOI 10.1007/s11464-009-0014-8 On vertex-coloring edge-weighting of graphs Hongliang LU 1, Xu YANG 1, Qinglin YU 1,2 1 Center for Combinatorics, Key Laboratory of Pure Mathematics and
More informationDO NOT RE-DISTRIBUTE THIS SOLUTION FILE
Professor Kindred Math 104, Graph Theory Homework 2 Solutions February 7, 2013 Introduction to Graph Theory, West Section 1.2: 26, 38, 42 Section 1.3: 14, 18 Section 2.1: 26, 29, 30 DO NOT RE-DISTRIBUTE
More informationCollapsible biclaw-free graphs
Collapsible biclaw-free graphs Hong-Jian Lai, Xiangjuan Yao February 24, 2006 Abstract A graph is called biclaw-free if it has no biclaw as an induced subgraph. In this note, we prove that if G is a connected
More informationMatching and Factor-Critical Property in 3-Dominating-Critical Graphs
Matching and Factor-Critical Property in 3-Dominating-Critical Graphs Tao Wang a,, Qinglin Yu a,b a Center for Combinatorics, LPMC Nankai University, Tianjin, China b Department of Mathematics and Statistics
More informationMAXIMUM WIENER INDEX OF TREES WITH GIVEN SEGMENT SEQUENCE
MAXIMUM WIENER INDEX OF TREES WITH GIVEN SEGMENT SEQUENCE ERIC OULD DADAH ANDRIANTIANA, STEPHAN WAGNER, AND HUA WANG Abstract. A segment of a tree is a path whose ends are branching vertices (vertices
More informationRestricted edge connectivity and restricted connectivity of graphs
Restricted edge connectivity and restricted connectivity of graphs Litao Guo School of Applied Mathematics Xiamen University of Technology Xiamen Fujian 361024 P.R.China ltguo2012@126.com Xiaofeng Guo
More informationThe Connectivity and Diameter of Second Order Circuit Graphs of Matroids
Graphs and Combinatorics (2012) 28:737 742 DOI 10.1007/s00373-011-1074-6 ORIGINAL PAPER The Connectivity and Diameter of Second Order Circuit Graphs of Matroids Jinquan Xu Ping Li Hong-Jian Lai Received:
More informationGEODETIC DOMINATION IN GRAPHS
GEODETIC DOMINATION IN GRAPHS H. Escuadro 1, R. Gera 2, A. Hansberg, N. Jafari Rad 4, and L. Volkmann 1 Department of Mathematics, Juniata College Huntingdon, PA 16652; escuadro@juniata.edu 2 Department
More informationComponent connectivity of crossed cubes
Component connectivity of crossed cubes School of Applied Mathematics Xiamen University of Technology Xiamen Fujian 361024 P.R.China ltguo2012@126.com Abstract: Let G = (V, E) be a connected graph. A r-component
More informationOn Geometric-Arithmetic Indices of (Molecular) Trees, Unicyclic Graphs and Bicyclic Graphs
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 66 (20) 68-697 ISSN 0340-623 On Geometric-Arithmetic Indices of (Molecular) Trees, Unicyclic Graphs and
More informationProperly Colored Paths and Cycles in Complete Graphs
011 ¼ 9 È È 15 ± 3 ¾ Sept., 011 Operations Research Transactions Vol.15 No.3 Properly Colored Paths and Cycles in Complete Graphs Wang Guanghui 1 ZHOU Shan Abstract Let K c n denote a complete graph on
More informationA note on isolate domination
Electronic Journal of Graph Theory and Applications 4 (1) (016), 94 100 A note on isolate domination I. Sahul Hamid a, S. Balamurugan b, A. Navaneethakrishnan c a Department of Mathematics, The Madura
More informationVertex-Colouring Edge-Weightings
Vertex-Colouring Edge-Weightings L. Addario-Berry a, K. Dalal a, C. McDiarmid b, B. A. Reed a and A. Thomason c a School of Computer Science, McGill University, University St. Montreal, QC, H3A A7, Canada
More informationA Note on Vertex Arboricity of Toroidal Graphs without 7-Cycles 1
International Mathematical Forum, Vol. 11, 016, no. 14, 679-686 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/imf.016.667 A Note on Vertex Arboricity of Toroidal Graphs without 7-Cycles 1 Haihui
More informationTriple Connected Domination Number of a Graph
International J.Math. Combin. Vol.3(2012), 93-104 Triple Connected Domination Number of a Graph G.Mahadevan, Selvam Avadayappan, J.Paulraj Joseph and T.Subramanian Department of Mathematics Anna University:
More informationProgress Towards the Total Domination Game 3 4 -Conjecture
Progress Towards the Total Domination Game 3 4 -Conjecture 1 Michael A. Henning and 2 Douglas F. Rall 1 Department of Pure and Applied Mathematics University of Johannesburg Auckland Park, 2006 South Africa
More informationModule 7. Independent sets, coverings. and matchings. Contents
Module 7 Independent sets, coverings Contents and matchings 7.1 Introduction.......................... 152 7.2 Independent sets and coverings: basic equations..... 152 7.3 Matchings in bipartite graphs................
More informationAdjacent Vertex Distinguishing Incidence Coloring of the Cartesian Product of Some Graphs
Journal of Mathematical Research & Exposition Mar., 2011, Vol. 31, No. 2, pp. 366 370 DOI:10.3770/j.issn:1000-341X.2011.02.022 Http://jmre.dlut.edu.cn Adjacent Vertex Distinguishing Incidence Coloring
More informationTHE INSULATION SEQUENCE OF A GRAPH
THE INSULATION SEQUENCE OF A GRAPH ELENA GRIGORESCU Abstract. In a graph G, a k-insulated set S is a subset of the vertices of G such that every vertex in S is adjacent to at most k vertices in S, and
More informationThe Edge Fixing Edge-To-Vertex Monophonic Number Of A Graph
Applied Mathematics E-Notes, 15(2015), 261-275 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ The Edge Fixing Edge-To-Vertex Monophonic Number Of A Graph KrishnaPillai
More information6. Lecture notes on matroid intersection
Massachusetts Institute of Technology 18.453: Combinatorial Optimization Michel X. Goemans May 2, 2017 6. Lecture notes on matroid intersection One nice feature about matroids is that a simple greedy algorithm
More informationGraph Connectivity G G G
Graph Connectivity 1 Introduction We have seen that trees are minimally connected graphs, i.e., deleting any edge of the tree gives us a disconnected graph. What makes trees so susceptible to edge deletions?
More informationMaximum number of edges in claw-free graphs whose maximum degree and matching number are bounded
Maximum number of edges in claw-free graphs whose maximum degree and matching number are bounded Cemil Dibek Tınaz Ekim Pinar Heggernes Abstract We determine the maximum number of edges that a claw-free
More informationK 4 C 5. Figure 4.5: Some well known family of graphs
08 CHAPTER. TOPICS IN CLASSICAL GRAPH THEORY K, K K K, K K, K K, K C C C C 6 6 P P P P P. Graph Operations Figure.: Some well known family of graphs A graph Y = (V,E ) is said to be a subgraph of a graph
More informationSkew propagation time
Graduate Theses and Dissertations Graduate College 015 Skew propagation time Nicole F. Kingsley Iowa State University Follow this and additional works at: http://lib.dr.iastate.edu/etd Part of the Applied
More informationDiscrete Applied Mathematics. A revision and extension of results on 4-regular, 4-connected, claw-free graphs
Discrete Applied Mathematics 159 (2011) 1225 1230 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam A revision and extension of results
More informationExtremal Graph Theory: Turán s Theorem
Bridgewater State University Virtual Commons - Bridgewater State University Honors Program Theses and Projects Undergraduate Honors Program 5-9-07 Extremal Graph Theory: Turán s Theorem Vincent Vascimini
More informationDefinition For vertices u, v V (G), the distance from u to v, denoted d(u, v), in G is the length of a shortest u, v-path. 1
Graph fundamentals Bipartite graph characterization Lemma. If a graph contains an odd closed walk, then it contains an odd cycle. Proof strategy: Consider a shortest closed odd walk W. If W is not a cycle,
More informationErdös-Gallai-type results for conflict-free connection of graphs
Erdös-Gallai-type results for conflict-free connection of graphs Meng Ji 1, Xueliang Li 1,2 1 Center for Combinatorics and LPMC arxiv:1812.10701v1 [math.co] 27 Dec 2018 Nankai University, Tianjin 300071,
More informationON WIENER INDEX OF GRAPH COMPLEMENTS. Communicated by Alireza Abdollahi. 1. Introduction
Transactions on Combinatorics ISSN (print): 51-8657, ISSN (on-line): 51-8665 Vol. 3 No. (014), pp. 11-15. c 014 University of Isfahan www.combinatorics.ir www.ui.ac.ir ON WIENER INDEX OF GRAPH COMPLEMENTS
More informationOn Geometric Arithmetic Index of Graphs
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 64 2010) 619-630 ISSN 0340-6253 On Geometric Arithmetic Index of Graphs Kinkar Ch. Das Department of Mathematics,
More informationVertex 3-colorability of claw-free graphs
Algorithmic Operations Research Vol.2 (27) 5 2 Vertex 3-colorability of claw-free graphs Marcin Kamiński a Vadim Lozin a a RUTCOR - Rutgers University Center for Operations Research, 64 Bartholomew Road,
More informationEDGE MAXIMAL GRAPHS CONTAINING NO SPECIFIC WHEELS. Jordan Journal of Mathematics and Statistics (JJMS) 8(2), 2015, pp I.
EDGE MAXIMAL GRAPHS CONTAINING NO SPECIFIC WHEELS M.S.A. BATAINEH (1), M.M.M. JARADAT (2) AND A.M.M. JARADAT (3) A. Let k 4 be a positive integer. Let G(n; W k ) denote the class of graphs on n vertices
More informationBipartite Roots of Graphs
Bipartite Roots of Graphs Lap Chi Lau Department of Computer Science University of Toronto Graph H is a root of graph G if there exists a positive integer k such that x and y are adjacent in G if and only
More informationAbstract. A graph G is perfect if for every induced subgraph H of G, the chromatic number of H is equal to the size of the largest clique of H.
Abstract We discuss a class of graphs called perfect graphs. After defining them and getting intuition with a few simple examples (and one less simple example), we present a proof of the Weak Perfect Graph
More informationColoring edges and vertices of graphs without short or long cycles
Coloring edges and vertices of graphs without short or long cycles Marcin Kamiński and Vadim Lozin Abstract Vertex and edge colorability are two graph problems that are NPhard in general. We show that
More informationCOLORING EDGES AND VERTICES OF GRAPHS WITHOUT SHORT OR LONG CYCLES
Volume 2, Number 1, Pages 61 66 ISSN 1715-0868 COLORING EDGES AND VERTICES OF GRAPHS WITHOUT SHORT OR LONG CYCLES MARCIN KAMIŃSKI AND VADIM LOZIN Abstract. Vertex and edge colorability are two graph problems
More informationBounds on the k-domination Number of a Graph
Bounds on the k-domination Number of a Graph Ermelinda DeLaViña a,1, Wayne Goddard b, Michael A. Henning c,, Ryan Pepper a,1, Emil R. Vaughan d a University of Houston Downtown b Clemson University c University
More informationRAINBOW CONNECTION AND STRONG RAINBOW CONNECTION NUMBERS OF
RAINBOW CONNECTION AND STRONG RAINBOW CONNECTION NUMBERS OF Srava Chrisdes Antoro Fakultas Ilmu Komputer, Universitas Gunadarma srava_chrisdes@staffgunadarmaacid Abstract A rainbow path in an edge coloring
More informationSubdivisions of Graphs: A Generalization of Paths and Cycles
Subdivisions of Graphs: A Generalization of Paths and Cycles Ch. Sobhan Babu and Ajit A. Diwan Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076,
More informationMinimum Cycle Bases of Halin Graphs
Minimum Cycle Bases of Halin Graphs Peter F. Stadler INSTITUTE FOR THEORETICAL CHEMISTRY AND MOLECULAR STRUCTURAL BIOLOGY, UNIVERSITY OF VIENNA WÄHRINGERSTRASSE 17, A-1090 VIENNA, AUSTRIA, & THE SANTA
More informationFundamental Properties of Graphs
Chapter three In many real-life situations we need to know how robust a graph that represents a certain network is, how edges or vertices can be removed without completely destroying the overall connectivity,
More informationSome Upper Bounds for Signed Star Domination Number of Graphs. S. Akbari, A. Norouzi-Fard, A. Rezaei, R. Rotabi, S. Sabour.
Some Upper Bounds for Signed Star Domination Number of Graphs S. Akbari, A. Norouzi-Fard, A. Rezaei, R. Rotabi, S. Sabour Abstract Let G be a graph with the vertex set V (G) and edge set E(G). A function
More informationNotes on Trees with Minimal Atom Bond Connectivity Index
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 67 (2012) 467-482 ISSN 0340-6253 Notes on Trees with Minimal Atom Bond Connectivity Index Ivan Gutman, Boris
More informationNumber Theory and Graph Theory
1 Number Theory and Graph Theory Chapter 6 Basic concepts and definitions of graph theory By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: satya8118@gmail.com
More informationDOUBLE DOMINATION CRITICAL AND STABLE GRAPHS UPON VERTEX REMOVAL 1
Discussiones Mathematicae Graph Theory 32 (2012) 643 657 doi:10.7151/dmgt.1633 DOUBLE DOMINATION CRITICAL AND STABLE GRAPHS UPON VERTEX REMOVAL 1 Soufiane Khelifi Laboratory LMP2M, Bloc of laboratories
More informationAdjacent: Two distinct vertices u, v are adjacent if there is an edge with ends u, v. In this case we let uv denote such an edge.
1 Graph Basics What is a graph? Graph: a graph G consists of a set of vertices, denoted V (G), a set of edges, denoted E(G), and a relation called incidence so that each edge is incident with either one
More informationVertex Colorings without Rainbow or Monochromatic Subgraphs. 1 Introduction
Vertex Colorings without Rainbow or Monochromatic Subgraphs Wayne Goddard and Honghai Xu Dept of Mathematical Sciences, Clemson University Clemson SC 29634 {goddard,honghax}@clemson.edu Abstract. This
More informationRoman Domination in Complementary Prism Graphs
International J.Math. Combin. Vol.2(2012), 24-31 Roman Domination in Complementary Prism Graphs B.Chaluvaraju and V.Chaitra 1(Department of Mathematics, Bangalore University, Central College Campus, Bangalore
More informationHW Graph Theory SOLUTIONS (hbovik)
Diestel 1.3: Let G be a graph containing a cycle C, and assume that G contains a path P of length at least k between two vertices of C. Show that G contains a cycle of length at least k. If C has length
More informationThe strong chromatic number of a graph
The strong chromatic number of a graph Noga Alon Abstract It is shown that there is an absolute constant c with the following property: For any two graphs G 1 = (V, E 1 ) and G 2 = (V, E 2 ) on the same
More informationTheorem 3.1 (Berge) A matching M in G is maximum if and only if there is no M- augmenting path.
3 Matchings Hall s Theorem Matching: A matching in G is a subset M E(G) so that no edge in M is a loop, and no two edges in M are incident with a common vertex. A matching M is maximal if there is no matching
More informationOn vertex types of graphs
On vertex types of graphs arxiv:1705.09540v1 [math.co] 26 May 2017 Pu Qiao, Xingzhi Zhan Department of Mathematics, East China Normal University, Shanghai 200241, China Abstract The vertices of a graph
More informationarxiv: v1 [math.co] 24 Oct 2012
On a relation between the Szeged index and the Wiener index for bipartite graphs arxiv:110.6460v1 [math.co] 4 Oct 01 Lily Chen, Xueliang Li, Mengmeng Liu Center for Combinatorics, LPMC-TJKLC Nankai University,
More informationPartitioning Complete Multipartite Graphs by Monochromatic Trees
Partitioning Complete Multipartite Graphs by Monochromatic Trees Atsushi Kaneko, M.Kano 1 and Kazuhiro Suzuki 1 1 Department of Computer and Information Sciences Ibaraki University, Hitachi 316-8511 Japan
More informationIndependence Number and Cut-Vertices
Independence Number and Cut-Vertices Ryan Pepper University of Houston Downtown, Houston, Texas 7700 pepperr@uhd.edu Abstract We show that for any connected graph G, α(g) C(G) +1, where α(g) is the independence
More informationThe Crossing Numbers of Join of a Subdivision of K 2,3 with P n and C n
Journal of Mathematical Research with Applications Nov., 2017, Vol.37, No.6, pp.649 656 DOI:10.3770/j.issn:2095-2651.2017.06.002 Http://jmre.dlut.edu.cn The Crossing Numbers of Join of a Subdivision of
More informationBounds for the m-eternal Domination Number of a Graph
Bounds for the m-eternal Domination Number of a Graph Michael A. Henning Department of Pure and Applied Mathematics University of Johannesburg South Africa mahenning@uj.ac.za Gary MacGillivray Department
More informationCLAW-FREE 3-CONNECTED P 11 -FREE GRAPHS ARE HAMILTONIAN
CLAW-FREE 3-CONNECTED P 11 -FREE GRAPHS ARE HAMILTONIAN TOMASZ LUCZAK AND FLORIAN PFENDER Abstract. We show that every 3-connected claw-free graph which contains no induced copy of P 11 is hamiltonian.
More informationProblem Set 2 Solutions
Problem Set 2 Solutions Graph Theory 2016 EPFL Frank de Zeeuw & Claudiu Valculescu 1. Prove that the following statements about a graph G are equivalent. - G is a tree; - G is minimally connected (it is
More informationTotal forcing number of the triangular grid
Mathematical Communications 9(2004), 169-179 169 Total forcing number of the triangular grid Damir Vukičević and Jelena Sedlar Abstract. LetT be a square triangular grid with n rows and columns of vertices
More informationThe Restrained Edge Geodetic Number of a Graph
International Journal of Computational and Applied Mathematics. ISSN 0973-1768 Volume 11, Number 1 (2016), pp. 9 19 Research India Publications http://www.ripublication.com/ijcam.htm The Restrained Edge
More informationA study on the Primitive Holes of Certain Graphs
A study on the Primitive Holes of Certain Graphs Johan Kok arxiv:150304526v1 [mathco] 16 Mar 2015 Tshwane Metropolitan Police Department City of Tshwane, Republic of South Africa E-mail: kokkiek2@tshwanegovza
More informationMC 302 GRAPH THEORY 10/1/13 Solutions to HW #2 50 points + 6 XC points
MC 0 GRAPH THEORY 0// Solutions to HW # 0 points + XC points ) [CH] p.,..7. This problem introduces an important class of graphs called the hypercubes or k-cubes, Q, Q, Q, etc. I suggest that before you
More information[8] that this cannot happen on the projective plane (cf. also [2]) and the results of Robertson, Seymour, and Thomas [5] on linkless embeddings of gra
Apex graphs with embeddings of face-width three Bojan Mohar Department of Mathematics University of Ljubljana Jadranska 19, 61111 Ljubljana Slovenia bojan.mohar@uni-lj.si Abstract Aa apex graph is a graph
More informationCharacterizing Graphs (3) Characterizing Graphs (1) Characterizing Graphs (2) Characterizing Graphs (4)
S-72.2420/T-79.5203 Basic Concepts 1 S-72.2420/T-79.5203 Basic Concepts 3 Characterizing Graphs (1) Characterizing Graphs (3) Characterizing a class G by a condition P means proving the equivalence G G
More informationLOWER BOUNDS FOR THE DOMINATION NUMBER
Discussiones Mathematicae Graph Theory 0 (010 ) 475 487 LOWER BOUNDS FOR THE DOMINATION NUMBER Ermelinda Delaviña, Ryan Pepper and Bill Waller University of Houston Downtown Houston, TX, 7700, USA Abstract
More informationTwo Characterizations of Hypercubes
Two Characterizations of Hypercubes Juhani Nieminen, Matti Peltola and Pasi Ruotsalainen Department of Mathematics, University of Oulu University of Oulu, Faculty of Technology, Mathematics Division, P.O.
More informationEdge Colorings of Complete Multipartite Graphs Forbidding Rainbow Cycles
Theory and Applications of Graphs Volume 4 Issue 2 Article 2 November 2017 Edge Colorings of Complete Multipartite Graphs Forbidding Rainbow Cycles Peter Johnson johnspd@auburn.edu Andrew Owens Auburn
More information{ 1} Definitions. 10. Extremal graph theory. Problem definition Paths and cycles Complete subgraphs
Problem definition Paths and cycles Complete subgraphs 10. Extremal graph theory 10.1. Definitions Let us examine the following forbidden subgraph problems: At most how many edges are in a graph of order
More informationModule 11. Directed Graphs. Contents
Module 11 Directed Graphs Contents 11.1 Basic concepts......................... 256 Underlying graph of a digraph................ 257 Out-degrees and in-degrees.................. 258 Isomorphism..........................
More information[Ramalingam, 4(12): December 2017] ISSN DOI /zenodo Impact Factor
GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES FORCING VERTEX TRIANGLE FREE DETOUR NUMBER OF A GRAPH S. Sethu Ramalingam * 1, I. Keerthi Asir 2 and S. Athisayanathan 3 *1,2 & 3 Department of Mathematics,
More informationOn the packing chromatic number of some lattices
On the packing chromatic number of some lattices Arthur S. Finbow Department of Mathematics and Computing Science Saint Mary s University Halifax, Canada BH C art.finbow@stmarys.ca Douglas F. Rall Department
More informationON THE EMPTY CONVEX PARTITION OF A FINITE SET IN THE PLANE**
Chin. Ann. of Math. 23B:(2002),87-9. ON THE EMPTY CONVEX PARTITION OF A FINITE SET IN THE PLANE** XU Changqing* DING Ren* Abstract The authors discuss the partition of a finite set of points in the plane
More informationA step towards the Bermond-Thomassen conjecture about disjoint cycles in digraphs
A step towards the Bermond-Thomassen conjecture about disjoint cycles in digraphs Nicolas Lichiardopol Attila Pór Jean-Sébastien Sereni Abstract In 1981, Bermond and Thomassen conjectured that every digraph
More informationThe Dual Neighborhood Number of a Graph
Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 47, 2327-2334 The Dual Neighborhood Number of a Graph B. Chaluvaraju 1, V. Lokesha 2 and C. Nandeesh Kumar 1 1 Department of Mathematics Central College
More informationBounds on distances for spanning trees of graphs. Mr Mthobisi Luca Ntuli
Bounds on distances for spanning trees of graphs Mr Mthobisi Luca Ntuli March 8, 2018 To Mphemba Legacy iii Acknowledgments I would like to thank my supervisors, Dr MJ Morgan and Prof S Mukwembi. It
More informationComplete Bipartite Graphs with No Rainbow Paths
International Journal of Contemporary Mathematical Sciences Vol. 11, 2016, no. 10, 455-462 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2016.6951 Complete Bipartite Graphs with No Rainbow
More informationPartitions and Packings of Complete Geometric Graphs with Plane Spanning Double Stars and Paths
Partitions and Packings of Complete Geometric Graphs with Plane Spanning Double Stars and Paths Master Thesis Patrick Schnider July 25, 2015 Advisors: Prof. Dr. Emo Welzl, Manuel Wettstein Department of
More informationComponent Connectivity of Generalized Petersen Graphs
March 11, 01 International Journal of Computer Mathematics FeHa0 11 01 To appear in the International Journal of Computer Mathematics Vol. 00, No. 00, Month 01X, 1 1 Component Connectivity of Generalized
More informationPACKING DIGRAPHS WITH DIRECTED CLOSED TRAILS
PACKING DIGRAPHS WITH DIRECTED CLOSED TRAILS PAUL BALISTER Abstract It has been shown [Balister, 2001] that if n is odd and m 1,, m t are integers with m i 3 and t i=1 m i = E(K n) then K n can be decomposed
More informationDisjoint directed cycles
Disjoint directed cycles Noga Alon Abstract It is shown that there exists a positive ɛ so that for any integer k, every directed graph with minimum outdegree at least k contains at least ɛk vertex disjoint
More informationMath 170- Graph Theory Notes
1 Math 170- Graph Theory Notes Michael Levet December 3, 2018 Notation: Let n be a positive integer. Denote [n] to be the set {1, 2,..., n}. So for example, [3] = {1, 2, 3}. To quote Bud Brown, Graph theory
More informationON THE NON-(p 1)-PARTITE K p -FREE GRAPHS
Discussiones Mathematicae Graph Theory 33 (013) 9 3 doi:10.7151/dmgt.1654 Dedicated to the 70th Birthday of Mieczys law Borowiecki ON THE NON-(p 1)-PARTITE K p -FREE GRAPHS Kinnari Amin Department of Mathematics,
More informationAverage D-distance Between Edges of a Graph
Indian Journal of Science and Technology, Vol 8(), 5 56, January 05 ISSN (Print) : 0974-6846 ISSN (Online) : 0974-5645 OI : 07485/ijst/05/v8i/58066 Average -distance Between Edges of a Graph Reddy Babu
More informationBinding Number of Some Special Classes of Trees
International J.Math. Combin. Vol.(206), 76-8 Binding Number of Some Special Classes of Trees B.Chaluvaraju, H.S.Boregowda 2 and S.Kumbinarsaiah 3 Department of Mathematics, Bangalore University, Janana
More informationStar Decompositions of the Complete Split Graph
University of Dayton ecommons Honors Theses University Honors Program 4-016 Star Decompositions of the Complete Split Graph Adam C. Volk Follow this and additional works at: https://ecommons.udayton.edu/uhp_theses
More informationSome Elementary Lower Bounds on the Matching Number of Bipartite Graphs
Some Elementary Lower Bounds on the Matching Number of Bipartite Graphs Ermelinda DeLaViña and Iride Gramajo Department of Computer and Mathematical Sciences University of Houston-Downtown Houston, Texas
More informationCycles through specified vertices in triangle-free graphs
March 6, 2006 Cycles through specified vertices in triangle-free graphs Daniel Paulusma Department of Computer Science, Durham University Science Laboratories, South Road, Durham DH1 3LE, England daniel.paulusma@durham.ac.uk
More informationVertex Magic Total Labelings of Complete Graphs 1
Vertex Magic Total Labelings of Complete Graphs 1 Krishnappa. H. K. and Kishore Kothapalli and V. Ch. Venkaiah Centre for Security, Theory, and Algorithmic Research International Institute of Information
More informationDO NOT RE-DISTRIBUTE THIS SOLUTION FILE
Professor Kindred Math 104, Graph Theory Homework 3 Solutions February 14, 2013 Introduction to Graph Theory, West Section 2.1: 37, 62 Section 2.2: 6, 7, 15 Section 2.3: 7, 10, 14 DO NOT RE-DISTRIBUTE
More informationEstrada Index. Bo Zhou. Augest 5, Department of Mathematics, South China Normal University
Outline 1. Introduction 2. Results for 3. References Bo Zhou Department of Mathematics, South China Normal University Augest 5, 2010 Outline 1. Introduction 2. Results for 3. References Outline 1. Introduction
More informationRainbow game domination subdivision number of a graph
Rainbow game domination subdivision number of a graph J. Amjadi Department of Mathematics Azarbaijan Shahid Madani University Tabriz, I.R. Iran j-amjadi@azaruniv.edu Abstract The rainbow game domination
More informationChapter 3: Paths and Cycles
Chapter 3: Paths and Cycles 5 Connectivity 1. Definitions: Walk: finite sequence of edges in which any two consecutive edges are adjacent or identical. (Initial vertex, Final vertex, length) Trail: walk
More informationDOMINATION GAME: EXTREMAL FAMILIES FOR THE 3/5-CONJECTURE FOR FORESTS
Discussiones Mathematicae Graph Theory 37 (2017) 369 381 doi:10.7151/dmgt.1931 DOMINATION GAME: EXTREMAL FAMILIES FOR THE 3/5-CONJECTURE FOR FORESTS Michael A. Henning 1 Department of Pure and Applied
More informationGraph Theory S 1 I 2 I 1 S 2 I 1 I 2
Graph Theory S I I S S I I S Graphs Definition A graph G is a pair consisting of a vertex set V (G), and an edge set E(G) ( ) V (G). x and y are the endpoints of edge e = {x, y}. They are called adjacent
More informationDecreasing the Diameter of Bounded Degree Graphs
Decreasing the Diameter of Bounded Degree Graphs Noga Alon András Gyárfás Miklós Ruszinkó February, 00 To the memory of Paul Erdős Abstract Let f d (G) denote the minimum number of edges that have to be
More information