On graphs of minimum skew rank 4
|
|
- Grant Mitchell
- 6 years ago
- Views:
Transcription
1 On graphs of minimum skew rank 4 Sudipta Mallik a and Bryan L. Shader b a Department of Mathematics & Statistics, Northern Arizona University, 85 S. Osborne Dr. PO Box: 5717, Flagstaff, AZ 8611, USA b Department of Mathematics, University of Wyoming, 1 E University Avenue, Laramie,WY 8271, USA Abstract The zero-nonzero pattern of a skew-symmetric matrix defines a graph. The minimum rank of all real skew-symmetric matrices with a given graph is studied. The probabilistic method is used to show that for sufficiently large n, there is a regular graph G of order n whose complement has girth at least 5 and for which every skew-symmetric matrix with graph G has rank at least 6. This settles an open problem. It is also proved that almost all graphs on n vertices do not allow a skew-symmetric matrix whose rank is less than n 25. AMS classification: 5C5; 15A3. Keywords: Skew-symmetric matrix, skew rank, regular graph, average minimum skew rank, probabilistic method. 1 Introduction We begin by establishing some notation that follows [4]. Throughout all matrices are real. An n n skew-symmetric matrix A determines a graph G(A that has vertex set {1, 2,..., n} and edge set {{i, j} : a i,j, 1 i < j n}. For a graph G, S (G denotes the set of all skew-symmetric Corresponding author. sudipta.mallik@nau.edu 1
2 matrices whose graph is G. The minimum skew rank of the graph G is the minimum rank of a matrix in S (G and is denoted by mr (G. The edge joining vertices u and v is simply denoted by uv. For a vertex v of G, N(v denotes the neighbourhood of v, i.e., the set of all vertices u such that uv is an edge. Two non-adjacent vertices u and v, u v, of a graph G are called duplicate vertices if N(u = N(v. The complete graph on n vertices is denoted by K n. The graph obtained from K 4 by deleting an edge is called a diamond. The complement graph of a diamond is a co-diamond. The union of k vertex disjoint copies of the graph H is denoted by kh. The graph 2K 3 + e is obtained from 2K 3 by inserting one additional edge (see Figure 1. Figure 1: co-diamond and 2K 3 + e The problem of characterizing graphs with minimum skew rank 4 is introduced in [3]. Deleting isolated vertices or duplicate vertices from a graph does not change the minimum skew rank of the graph. The following necessary conditions for a graph G of order n 6 with neither duplicate vertices nor isolated vertices to have minimum skew rank at most 4 are given in [4]. If H is not an induced subgraph of a graph G, then G is H-free. Theorem 1.1. [4, Thm 2.6] Let G be a graph of order n 6 with neither duplicate vertices nor isolated vertices. If mr (G 4, then G is co-diamondfree and (2K 3 + e-free. It has been shown in [4] that these conditions are sufficient for several families of graphs, e.g., the complement of trees. In Section 2, we investigate the sufficiency of these conditions and show sufficiency for graphs of order n 8, and all but one graph of order 9. In Section 3, we prove that for n sufficiently large these conditions are far from sufficient. In Section 4, we give a lower bound on the average minimum skew rank of graphs of order n. A matching of a graph G is a set of edges of G where no two edges share a common vertex. A perfect matching of a graph G of order 2k is a matching consisting of k edges. Graphs of order n whose minimum skew rank is n are characterized as follows. 2
3 Theorem 1.2. [3, Theorem 2.6] Let G be a graph of order n. Then mr (G = n if and only if G has a unique perfect matching. We note that Theorem 1.2 implies that a graph of odd order n has minimum skew rank at most n 1. A cycle on n vertices is denoted by C n. The girth of a graph G is the length of a shortest cycle in G if G contains a cycle, and is infinity otherwise. For example, a graph G with girth at least 5 is C 3 -free and C 4 -free. 2 Necessary conditions for graphs to have minimum skew rank at most 4 It is natural to ask if the converse of Theorem 1.1 is true. In this section we study this question for graphs of small order. It is often more convenient to study the equivalent problem: If mr (G 6 and G has neither duplicate vertices nor isolated vertices, then does G have a diamond or 2K 3 + e as an induced subgraph? First we show that the converse of Theorem 1.1 is true for disconnected graphs by using the following lemma. Lemma 2.1. [3, Obs 1.6, Thm 2.1] (a mr (K n = 2 for all n 2. (b If the connected components of G are G 1,..., G t, then mr (G = t mr (G i. Theorem 2.2. Let G be a disconnected graph of order n 6 with neither duplicate vertices nor isolated vertices. If G is co-diamond-free and (2K 3 +e- free, then G is a disjoint union of two complete graphs and mr (G = 4. Proof. Suppose that G is co-diamond-free, (2K 3 + e-free, and disconnected graph of order n 6 with neither duplicate vertices nor isolated vertices. Since G has no isolated vertices, each connected component of G has at 3
4 least one edge. If G has at least three connected components, then G has an induced co-diamond. Thus G has exactly two connected components, each containing at least one edge. If any connected component contains two non-adjacent vertices, then these two vertices along with an edge of the other connected component will form a co-diamond in G. Thus each of two connected components of G is a complete graph. By Lemma 2.1, we get mr (G = 4. Now we verify the converse of Theorem 1.1 for connected graphs of small order. Let us start with connected graphs on 6 vertices. Suppose that connected graph G has 6 vertices with mr (G = 6 and with no duplicate vertices. By Theorem 1.2, G has a unique perfect matching. If G has a pendant edge uv with pendant vertex v, then the graph obtained from G by deleting u and v is a graph on 4 vertices with a perfect matching. It can be verified that G is a graph of the form of one of the graphs of Figure 4 other than 2K 3 + e. Now assume that G has minimum degree at least 2. Then G has at least three edges different from three edges of the perfect matching. Since G has a unique perfect matching and minimum degree at least 2, after appending two more edges to two vertices of an edge, say 12 of the perfect matching, G would have a subgraph of the form of one of the graphs of Figure Figure 2: Possible subgraphs of G on 6 vertices with 5 edges Since G has a unique perfect matching and the degree of vertex 4 is at least 2, after appending an edge to vertex 4, G would have a subgraph of the form of one of the graphs of Figure Figure 3: Possible subgraphs of G on 6 vertices with 6 edges 4
5 Since G has a unique perfect matching and the degree of vertex 5 is at least 2, after appending an edge to vertex 5, G is 2K 3 + e. Now it can be checked that except 2K 3 + e each of the graphs in Figure 4 has an induced co-diamond. Thus the converse of Theorem 1.1 holds for graphs on 6 vertices. any but at least 1 any but at least 1 Figure 4: Connected graphs on 6 vertices with a unique perfect matching Next we study the converse of Theorem 1.1 for connected graphs on more than 6 vertices using a technique from [4]. The matrix N is a basis matrix of the left null space of a matrix A if the rows of N form a basis of the left null space of A. A collection {N i : i I} of vectors is a minimal dependent set of vectors if it is a linearly dependent set of vectors and for each j I, {N i : i j, i I} is a linearly independent set of vectors. We denote the support of a row or column vector v R n by Z(v and define by Z(v = {i : 1 i n, v i }. Theorem 2.3. [4, Thm 4.8] Let A be a real skew-symmetric matrix of order n 2. Let N be a basis matrix of the left null space of A. If Z is a union of indices of minimal dependent columns of N, then there exists a vector w with support Z such that ([ ] A w rank w T = rank(a. Corollary 2.4. [4, Cor 4.9] Let G be a graph on n vertices. Let A S (G with rank(a = mr (G and N be a basis matrix of the left null space of A. Let G + v be a graph obtained by adjoining a new vertex v and some edges from v to G. If N(v is a union of indices of minimal dependent columns of N, then mr (G + v = mr (G. Enabled by Sage [8] and Brendan McKay s lists [6] of non-isomorphic connected graphs of small order, we are able to use Corollary 2.4 to show that for n 8, the necessary conditions in Theorem 1.1 are also sufficient. More precisely, for n 5, all graphs of order n have minimum skew rank at most 4. As argued earlier in this section, each graph of order 6 that has no 5
6 duplicate or isolated vertices, and no induced co-diamond or 2K 3 + e, has minimum skew rank at most 4. Next consider graphs of order 7. Starting with McKay s list, G(6, of connected graphs on 6 vertices, we eliminate all those that have a duplicate vertex, a co-diamond or a 2K 3 + e, to obtain a list Ĝ(6, which we know to have all connected graphs on 6 vertices minimum skew rank at most 4. For each H in Ĝ(6 we find (by hand a matrix A H in S (H of rank at most 4. These graphs and matrices are available at [8]. Starting with McKay s list, G(7, of connected graphs on 7 vertices, we eliminate all those that have a duplicate vertex, a co-diamond or a 2K 3 + e, to obtain a list Ĝ(7, which we know contains all counterexamples of order 7, if any exist. For each H in Ĝ(6, we use Corollary 2.4 to construct all graphs of order 7 which contain H as an induced subgraph and have a rank 4 realization whose leading 6 by 6 matrix is A H. Our Sage-supported computations show that all graphs in Ĝ(7 arise in this way, and hence the necessary conditions are also sufficient when n = 7. As we use Theorem 2.3 to build a graph G in Ĝ(7, we also find an A G in S (G with rank at most 4. These graphs and matrices are available at [8]. We move to graphs of order 8, and proceed in a similar manner. Our Sage-supported computations show that all graphs in Ĝ(8 arise this way, and hence the necessary conditions are also sufficient when n = 8. These graphs and matrices are available at [8]. In summary, we have shown the following. Theorem 2.5. Let G be a graph of order at most 8 with neither duplicate vertices nor isolated vertices. If G is co-diamond-free and (2K 3 + e-free, then mr (G 4. For graphs of order 9, we follow the similar procedure as before and we get two potential counterexamples which are shown in Figure 5, and are HEhbtjK and HEjbvj[ in graph6 format (see data/formats.html for details on this format. We note that the first graph is isomorphic to the Cartesian product K 3 K 3. 6
7 Figure 5: HEjbvj[ and HEhbtjK It can be verified that the skew-symmetric matrix has graph HEjbvj[ and rank 4. Hence mr (HEjbvj[ = 4. We now show that K 3 K 3 is indeed a counterexample. The adjacency matrix of K 3 K 3 is Note that this adjacency matrix is (J 3 I 3 (J 3 I 3, the Kronecker product of J 3 I 3 with itself where J 3 and I 3 are the all ones matrix and the identity matrix of order 3. Since K 3 K 3 has P 4 as an induced subgraph (on vertices 7.
8 2,3,4 and 5 in Figure 5, K 3 K 3 has minimum skew rank at least 4. Suppose that there exists a B S (K 3 K 3 of rank 4. Then columns 3i+1, 3i+2 and 3i + 3 are minimally linearly dependent for i =, 1, 2. Hence the nullspace of B contains vectors of the form a b c, d e f, and for some a, b, c, d, e, f, g, h, i each of which is nonzero. By replacing B by DBD where D is the diagonal matrix with main diagonal (a, b, c, d, e, f, g, h, i matrix, we may assume without loss of generality that a = b = c = d = e = f = g = h = i = 1. Thus, B has the form: x x y y x x y y x x y y x x z z x x z z x x z z y y z z y y z z y y z z for some nonzero x, y, z. Calculation shows that det B[{1, 2, 4, 5, 7, 8}] = 4x 2 y 2 z 2, which contradicts the assumption that B has rank 4. Hence mr (K 3 K 3 6. Thus, we have proven the following. Theorem 2.6. Let G be a graph of order at most 9 with neither duplicate vertices nor isolated vertices. If G is co-diamond-free and (2K 3 +e-free, and G is not isomorphic to K 3 K 3 then mr (G 4. g h i 8
9 3 Existence of co-diamond-free and (2K 3 + e- free graphs with minimum skew rank greater than 4 In this section we show that the converse of Theorem 1.1 is far from true for n sufficiently large. The technique used is a common technique in combinatorics, known as the probabilistic method (see [1]. This method is a non-constructive method that proves the existence (or ubiquity of mathematical object by showing that if one selects the object from a specified class the probability the the object has the desired property is greater than. We will use the following lemma to give an upper bound on the number of labelled graphs of order v with minimum skew rank at most 2k. Recall from Section 2 that the support of a row or column vector v R n is Z(v = {i : 1 i n, v i }. If f = (f 1,..., f m is a sequence of polynomials in n variables over the field F, then the zero-pattern set of f is {Z(f 1 (x,..., f m (x : x F n } and the cardinality of this set is denoted by Z F (f. Throughout the remainder of this article e is the base of the natural logarithm. Lemma 3.1. [7, Cor. 1.5] Let f = (f 1,..., f m be a sequence of polynomials of degree at most d in n variables over the field F, where d 1. Then ( n emd Z F (f <. n We now explain how the graphs of order n with minimum skew rank at most 2k corresponds to the zero-pattern set of a family of polynomials. First note that if A is a real skew-symmetric matrix of order v with rank at most 2k, then there exist v k matrices X and Ŷ such that A = XŶ T Ŷ X T [3, Lemma 1.3]. Now let x i,j and y i,j be distinct variables for 1 i v and 1 j k and let X = [x i,j ] and Y = [y i,j ]. Set p i,j = (XY T Y X T i,j. Consider the m-tuple f = (p i,j : 1 i < j v of polynomials in n variables over R where m = v(v 1/2 and n = 2vk. Now each zero pattern of f obtained by p i,j = a i,j, 1 i < j v corresponds to a skew-symmetric matrix A = [a i,j ] of order v and rank at most 2k whose graph G has mr (G 2k. Also each graph G of order v with mr (G 2k has a skew-symmetric matrix of rank at most 2k whose zero pattern corresponds to a unique zero pattern of f. Thus Z R (f is the number of labelled graphs of order v with minimum skew rank at most 2k. Lemma 3.1 now implies the following. 9
10 Theorem 3.2. The number of labelled graphs of order v with minimum skew ( 2vk e(v 1 rank at most 2k is less than. 2k Corollary 3.3. The number ( of labelled graphs of order n with minimum skew 4n e(n 1 rank at most 4 is less than. 4 Now we identify a family of co-diamond-free and (2K 3 + e-free graphs without duplicate vertices. Proposition 3.4. Let H be a (k + 1-regular graph of order n with girth at least 5 where 2 k n 3. Let G be the complement of H. Then G is a co-diamond-free and (2K 3 + e-free graph with neither duplicate vertices nor isolated vertices. Proof. Since H has girth at least 5, H is C 4 -free and C 3 -free. Consequently G is (2K 3 + e-free and co-diamond-free. Since n k 2 1 and G is (n k 2-regular graph, G has no isolated vertices. We will show that G has no duplicate vertices. Suppose to the contrary that u and v are duplicate vertices of G. Since H is C 3 -free, each vertex w of G other than u and v is adjacent to both u and v in G. But then the degree of u in G is n 2 which contradicts the fact that G is (n k 2-regular graph with 2 k n 3. The following lemmas lead us to a lower bound of the number of certain regular labelled graphs of order n with girth greater than 4. Throughout our calculations we denote log 2 by lg. Lemma 3.5. [5, Corollary 2] For (d 1 2g 1 = o(n, the number of d-regular labelled graphs of order n with girth greater than g is ( g (d 1 i (nd! exp + o(1 2i as n. It is well known that (nd/2!2 nd/2 (d! n t ln(t t + 1 ln(t! t ln(t 1
11 for all t >. This implies that for positive integers k we have lg(k! k lg k and (3.1 lg((k 9 + k 8! > (k 9 + k 8 lg(k 9 + k 8 (k9 + k 8 (3.2 ln 2 Corollary 3.6. Let k be a positive integer and n = k 8. For n sufficiently large, the number of (1 + k-regular labelled graphs of order n with girth greater than 4 is at least n n9/8 3. Proof. Set g = 4 and d = 1 + k. Then (d 1 2g 1 = n 7 8 = o(n. Let N be the number of (1 + k-regular graphs of order n with girth greater than 4. By Lemma 3.5, there exist constants c k such that ( 4 (k 9 + k 8 k i! exp 2i + c k N = ((k 9 + k 8 /2!2 (k9 +k 8 /2 ((k + 1! k8 and lim k c k =. Taking lg of both sides we get lg N = lg((k 9 + k 8! ( 4 k i lg e + c k lg e 2i lg(((k 9 + k 8 /2! (k 9 + k 8 /2 k 8 lg((k + 1! Using (3.1 and (3.2 in the preceding equation, we get ( 4 lg N > (k 9 + k 8 lg(k 9 + k 8 (k9 + k 8 k i lg e + c k lg e ln 2 2i ( k9 + k 8 k 9 + k 8 lg k9 + k 8 k 8 (k + 1 lg(k ( 4 = (k 9 + k 8 lg(k 9 + k 8 (2 + ln 2(k9 + k 8 k i lg e + c k lg e 2 ln 2 2i ( k9 + k 8 k 9 + k 8 lg (k 9 + k 8 lg(k + 1. (
12 ( Since k9 +k 8 k lg 9 +k 8 > k9 +k 8 lg (k 9 + k 8, from (3.3 we get lg N > k9 + k 8 Since k9 +k 8 2 lg(k 9 + k 8 (2 + ln 2(k9 + k 8 2 ln 2 ( 4 k i lg e + c k lg e 2i (k 9 + k 8 lg(k + 1. (3.4 > k9 2 2, from (3.4 we get lg N > k9 2 lg(k9 + k 8 (2 + ln 2k9 ln 2 ( 4 k i lg e + c k lg e 2i (k 9 + k 8 lg(k + 1. (3.5 ( 4 k i Since lg e = 3k8 + 4k 3 + 6k k lg e > 2k 8, from (3.5 we 2i 24 get lg N > k9 2 lg(k9 + k 8 (2 + ln 2k9 2k 8 + c k lg e (k 9 + k 8 lg(k + 1 ln 2 = k9 ( lg(k 8 + lg(k + 1 (2 + ln 2k9 2k 8 + c k lg e (k 9 + k 8 lg(k ( ln 2 = k9 lg(k 8 2(2 + ln 2 lg(k + 1 2k 8 + c k lg e k 8 lg(k + 1. (3.6 2 ln 2 Thus So for r > 2, say r = 3, lg N k 9 lg(k 8 /r = r + terms that go to, as k. 2 Then lg N > k 9 lg(k 8 /3 for k sufficiently large. lg N > n 9/8 lg(n/3 for n sufficiently large. Hence N is at least 2 n9/8 lg n 3 = (2 lg n n9/8 3 = n n9/8 3 for sufficiently large n. Now we show that for n sufficiently large, the converse of Theorem 1.1 fails to hold. 12
13 Theorem 3.7. For n sufficiently large there exists a co-diamond-free and (2K 3 + e-free graph of order n with neither duplicate vertices nor isolated vertices and minimum skew rank at least 6. Proof. By Corollary 3.3, the number of labelled graphs of order n with minimum skew rank at most 4 is less than n 4n. Now by Corollary 3.6, the number of (1 + k-regular labelled graphs of order n = k 8 with girth greater than 4 is at least n n9/8 3 for sufficiently large n. Since n 4n < n n9/8 3 as n, for sufficiently large n there exists a (k + 1-regular graph H of order n = k 8 with girth at least 5 whose complement G has minimum skew rank at least 6. By Proposition 3.4, G is co-diamond-free and (2K 3 + e-free graph with neither duplicate vertices nor isolated vertices. 4 Average minimum skew rank The average minimum rank of graphs of order v is introduced and studied in [2]. The average minimum skew rank of graphs of order v is denoted by amr (v and defined by amr (v = G mr (G, 2 (v 2 where the sum runs over all labelled graphs of order v. Thus, amr (v is the expected minimum skew rank of all labelled graphs on v vertices. In Theorem 3.2 we obtained an upper bound of the number of labelled graphs of order v with minimum skew rank at most 2k. This leads us to the following lower bound of the average minimum skew rank of graphs of order v. Theorem 4.1. Asymptotically almost all graphs on v vertices have minimum skew rank at least.4v. Proof. If N(v, 2k is the number of labelled graphs of order v with minimum skew rank at most 2k, then by Theorem 3.2, Thus N(v, 2k 2 (v 2 < 1 2 (v 2 ( 2vk e(v 1 N(v, 2k <. 2k ( 2vk ( e(v 1 e(v 1 = 2k 13 2k2 (v 1/(4k 2vk (4.1
14 Set k = v/5. Then from (4.1 we get N(v, v/25 2 (v 2 ( e(v 1 < 2k2 (v 1/(4k 2vk ( 25e(v 1 = v2 25(v 1/(2v v 2 25 = ( 25e(1 1/v 2 25(1 1/v/2 v 2 25 (4.2 25e(1 1/v Note that < 25e <.9 for v 2. Thus from (4.2 we get 2 25(1 1/v/2 225/4 N(v, v/25 as v. 2 (v 2 Corollary 4.2. For v sufficiently large, amr (v.4v. References [1] Noga Alon and Joel H. Spencer, The probabilistic method (2ed, New York: Wiley-Interscience, (2. [2] H. Tracy Hall, Leslie Hogben, Ryan Martin and Bryan Shader, Expected values of parameters associated with the minimum rank of a graph, Linear Algebra Appl. 433 ( [3] IMA-ISU research group on minimum rank (Mary Allison, Elizabeth Bodine, Luz Maria DeAlba, Joyati Debnath, Laura DeLoss, Colin Garnett, Jason Grout, Leslie Hogben, Bokhee Im, Hana Kim, Reshmi Nair, Olga Pryporova, Kendrick Savage, Bryan Shader, Amy Wangsness Wehe, Minimum rank of skew-symmetric matrices described by a graph, Linear Algebra Appl. 432 ( [4] Sudipta Mallik and Bryan L. Shader, Classes of graphs with minimum skew rank 4, Linear Algebra Appl. 439 ( [5] Brendan D. McKay, Nicholas C. Wormald and Beata Wysocka, Short cycles in random regular graphs, The Electronic Journal of Combinatorics 11 (24, # R66 [6] Brendan D. McKay, Collection of graphs as strings, [7] L. Rónyai, L. Babai, M.K. Ganapathy, On the number of zero-patterns of a sequence of polynomials, J. Amer. Math. Soc. 14 (3 ( (electronic. 14
15 [8] Minimum skew-rank of small order graphs website: edu/bshader/skew 15
Skew 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 informationLine Graphs and Circulants
Line Graphs and Circulants Jason Brown and Richard Hoshino Department of Mathematics and Statistics Dalhousie University Halifax, Nova Scotia, Canada B3H 3J5 Abstract The line graph of G, denoted L(G),
More informationREGULAR GRAPHS OF GIVEN GIRTH. Contents
REGULAR GRAPHS OF GIVEN GIRTH BROOKE ULLERY Contents 1. Introduction This paper gives an introduction to the area of graph theory dealing with properties of regular graphs of given girth. A large portion
More informationChapter 2. Splitting Operation and n-connected Matroids. 2.1 Introduction
Chapter 2 Splitting Operation and n-connected Matroids The splitting operation on an n-connected binary matroid may not yield an n-connected binary matroid. In this chapter, we provide a necessary and
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 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 informationOrthogonal representations, minimum rank, and graph complements
Orthogonal representations, minimum rank, and graph complements Leslie Hogben November 24, 2007 Abstract Orthogonal representations are used to show that complements of certain sparse graphs have (positive
More informationMath 776 Graph Theory Lecture Note 1 Basic concepts
Math 776 Graph Theory Lecture Note 1 Basic concepts Lectured by Lincoln Lu Transcribed by Lincoln Lu Graph theory was founded by the great Swiss mathematician Leonhard Euler (1707-178) after he solved
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 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 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 informationOrthogonal representations, minimum rank, and graph complements
Orthogonal representations, minimum rank, and graph complements Leslie Hogben March 30, 2007 Abstract Orthogonal representations are used to show that complements of certain sparse graphs have (positive
More informationMath 778S Spectral Graph Theory Handout #2: Basic graph theory
Math 778S Spectral Graph Theory Handout #: Basic graph theory Graph theory was founded by the great Swiss mathematician Leonhard Euler (1707-178) after he solved the Königsberg Bridge problem: Is it possible
More informationTitle Edge-minimal graphs of exponent 2
Provided by the author(s) and NUI Galway in accordance with publisher policies. Please cite the published version when available. Title Edge-minimal graphs of exponent 2 Author(s) O'Mahony, Olga Publication
More informationThe Structure of Bull-Free Perfect Graphs
The Structure of Bull-Free Perfect Graphs Maria Chudnovsky and Irena Penev Columbia University, New York, NY 10027 USA May 18, 2012 Abstract The bull is a graph consisting of a triangle and two vertex-disjoint
More informationMatching Algorithms. Proof. If a bipartite graph has a perfect matching, then it is easy to see that the right hand side is a necessary condition.
18.433 Combinatorial Optimization Matching Algorithms September 9,14,16 Lecturer: Santosh Vempala Given a graph G = (V, E), a matching M is a set of edges with the property that no two of the edges have
More informationA NOTE ON THE ASSOCIATED PRIMES OF THE THIRD POWER OF THE COVER IDEAL
A NOTE ON THE ASSOCIATED PRIMES OF THE THIRD POWER OF THE COVER IDEAL KIM KESTING, JAMES POZZI, AND JANET STRIULI Abstract. An algebraic approach to graph theory involves the study of the edge ideal and
More informationWinning Positions in Simplicial Nim
Winning Positions in Simplicial Nim David Horrocks Department of Mathematics and Statistics University of Prince Edward Island Charlottetown, Prince Edward Island, Canada, C1A 4P3 dhorrocks@upei.ca Submitted:
More informationc 2004 Society for Industrial and Applied Mathematics
SIAM J. MATRIX ANAL. APPL. Vol. 26, No. 2, pp. 390 399 c 2004 Society for Industrial and Applied Mathematics HERMITIAN MATRICES, EIGENVALUE MULTIPLICITIES, AND EIGENVECTOR COMPONENTS CHARLES R. JOHNSON
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 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 informationEdge-minimal graphs of exponent 2
JID:LAA AID:14042 /FLA [m1l; v1.204; Prn:24/02/2017; 12:28] P.1 (1-18) Linear Algebra and its Applications ( ) Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa
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 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 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 informationInfinite locally random graphs
Infinite locally random graphs Pierre Charbit and Alex D. Scott Abstract Motivated by copying models of the web graph, Bonato and Janssen [3] introduced the following simple construction: given a graph
More informationarxiv: v1 [math.co] 2 Jun 2017
arxiv:1706.00798v1 [math.co] Jun 017 Zero forcing number, Grundy domination number, and their variants Jephian C.-H. Lin June 6, 017 Abstract This paper presents strong connections between four variants
More informationA generalization of zero divisor graphs associated to commutative rings
Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:9 https://doi.org/10.1007/s12044-018-0389-0 A generalization of zero divisor graphs associated to commutative rings M. AFKHAMI 1, A. ERFANIAN 2,, K. KHASHYARMANESH
More informationThe Probabilistic Method
The Probabilistic Method Po-Shen Loh June 2010 1 Warm-up 1. (Russia 1996/4 In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees
More information1 Some Solution of Homework
Math 3116 Dr. Franz Rothe May 30, 2012 08SUM\3116_2012h1.tex Name: Use the back pages for extra space 1 Some Solution of Homework Proposition 1 (Counting labeled trees). There are n n 2 different labeled
More informationRigidity, connectivity and graph decompositions
First Prev Next Last Rigidity, connectivity and graph decompositions Brigitte Servatius Herman Servatius Worcester Polytechnic Institute Page 1 of 100 First Prev Next Last Page 2 of 100 We say that a framework
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 informationChapter 4. square sum graphs. 4.1 Introduction
Chapter 4 square sum graphs In this Chapter we introduce a new type of labeling of graphs which is closely related to the Diophantine Equation x 2 + y 2 = n and report results of our preliminary investigations
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 informationDiscrete Mathematics. Elixir Dis. Math. 92 (2016)
38758 Available online at www.elixirpublishers.com (Elixir International Journal) Discrete Mathematics Elixir Dis. Math. 92 (2016) 38758-38763 Complement of the Boolean Function Graph B(K p, INC, K q )
More informationOn Sequential Topogenic Graphs
Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 36, 1799-1805 On Sequential Topogenic Graphs Bindhu K. Thomas, K. A. Germina and Jisha Elizabath Joy Research Center & PG Department of Mathematics Mary
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 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 informationZero forcing number: Results for computation and comparison with other graph. parameters. Darren Daniel Row
Zero forcing number: Results for computation and comparison with other graph parameters by Darren Daniel Row A dissertation submitted to the graduate faculty in partial fulfillment of the requirements
More informationSmall Survey on Perfect Graphs
Small Survey on Perfect Graphs Michele Alberti ENS Lyon December 8, 2010 Abstract This is a small survey on the exciting world of Perfect Graphs. We will see when a graph is perfect and which are families
More informationON THE STRONGLY REGULAR GRAPH OF PARAMETERS
ON THE STRONGLY REGULAR GRAPH OF PARAMETERS (99, 14, 1, 2) SUZY LOU AND MAX MURIN Abstract. In an attempt to find a strongly regular graph of parameters (99, 14, 1, 2) or to disprove its existence, we
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 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 informationMinimum rank, maximum nullity and zero forcing number for selected graph families
Mathematics Publications Mathematics 2010 Minimum rank, maximum nullity and zero forcing number for selected graph families Edgard Almodovar University of Puerto Rico, Río Piedras Campus Laura DeLoss Iowa
More informationarxiv: v4 [math.co] 25 Apr 2010
QUIVERS OF FINITE MUTATION TYPE AND SKEW-SYMMETRIC MATRICES arxiv:0905.3613v4 [math.co] 25 Apr 2010 AHMET I. SEVEN Abstract. Quivers of finite mutation type are certain directed graphs that first arised
More informationDischarging and reducible configurations
Discharging and reducible configurations Zdeněk Dvořák March 24, 2018 Suppose we want to show that graphs from some hereditary class G are k- colorable. Clearly, we can restrict our attention to graphs
More informationDefinition: A graph G = (V, E) is called a tree if G is connected and acyclic. The following theorem captures many important facts about trees.
Tree 1. Trees and their Properties. Spanning trees 3. Minimum Spanning Trees 4. Applications of Minimum Spanning Trees 5. Minimum Spanning Tree Algorithms 1.1 Properties of Trees: Definition: A graph G
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 informationCHARACTERIZING SYMMETRIC DIAMETRICAL GRAPHS OF ORDER 12 AND DIAMETER 4
IJMMS 30:3 (2002) 145 149 PII. S0161171202012474 http://ijmms.hindawi.com Hindawi Publishing Corp. CHARACTERIZING SYMMETRIC DIAMETRICAL GRAPHS OF ORDER 12 AND DIAMETER 4 S. AL-ADDASI and H. Al-EZEH Received
More informationSubdivided graphs have linear Ramsey numbers
Subdivided graphs have linear Ramsey numbers Noga Alon Bellcore, Morristown, NJ 07960, USA and Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv,
More informationOn Rainbow Cycles in Edge Colored Complete Graphs. S. Akbari, O. Etesami, H. Mahini, M. Mahmoody. Abstract
On Rainbow Cycles in Edge Colored Complete Graphs S. Akbari, O. Etesami, H. Mahini, M. Mahmoody Abstract In this paper we consider optimal edge colored complete graphs. We show that in any optimal edge
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 informationExercise set 2 Solutions
Exercise set 2 Solutions Let H and H be the two components of T e and let F E(T ) consist of the edges of T with one endpoint in V (H), the other in V (H ) Since T is connected, F Furthermore, since T
More informationCHAPTER 2. Graphs. 1. Introduction to Graphs and Graph Isomorphism
CHAPTER 2 Graphs 1. Introduction to Graphs and Graph Isomorphism 1.1. The Graph Menagerie. Definition 1.1.1. A simple graph G = (V, E) consists of a set V of vertices and a set E of edges, represented
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 informationPart II. Graph Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 53 Paper 3, Section II 15H Define the Ramsey numbers R(s, t) for integers s, t 2. Show that R(s, t) exists for all s,
More informationAcyclic Edge Colorings of Graphs
Acyclic Edge Colorings of Graphs Noga Alon Ayal Zaks Abstract A proper coloring of the edges of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G,
More informationMatching Theory. Figure 1: Is this graph bipartite?
Matching Theory 1 Introduction A matching M of a graph is a subset of E such that no two edges in M share a vertex; edges which have this property are called independent edges. A matching M is said to
More informationAMS /672: Graph Theory Homework Problems - Week V. Problems to be handed in on Wednesday, March 2: 6, 8, 9, 11, 12.
AMS 550.47/67: Graph Theory Homework Problems - Week V Problems to be handed in on Wednesday, March : 6, 8, 9,,.. Assignment Problem. Suppose we have a set {J, J,..., J r } of r jobs to be filled by a
More informationOuter-2-independent domination in graphs
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, No. 1, February 2016, pp. 11 20. c Indian Academy of Sciences Outer-2-independent domination in graphs MARCIN KRZYWKOWSKI 1,2,, DOOST ALI MOJDEH 3 and MARYEM
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 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 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 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 informationON SWELL COLORED COMPLETE GRAPHS
Acta Math. Univ. Comenianae Vol. LXIII, (1994), pp. 303 308 303 ON SWELL COLORED COMPLETE GRAPHS C. WARD and S. SZABÓ Abstract. An edge-colored graph is said to be swell-colored if each triangle contains
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 informationarxiv: v3 [cs.dm] 24 Jul 2018
Equimatchable Claw-Free Graphs aieed Akbari a,1, Hadi Alizadeh b, Tınaz Ekim c, Didem Gözüpek b, Mordechai halom c,d,2 a Department of Mathematical ciences, harif University of Technology, 11155-9415,
More informationOn the Greedoid Polynomial for Rooted Graphs and Rooted Digraphs
On the Greedoid Polynomial for Rooted Graphs and Rooted Digraphs Elizabeth W. McMahon LAFAYETTE COLLEGE EASTON, PENNSYLVANIA ABSTRACT We examine some properties of the 2-variable greedoid polynomial f(g;t,
More informationON THE δ-conjecture FOR GRAPHS WITH MINIMUM DEGREE G 4. Matthew Villanueva
ON THE δ-conjecture FOR GRAPHS WITH MINIMUM DEGREE G 4 By Matthew Villanueva Francesco Barioli Associate Professor of Mathematics (Committee Chair) Stephen Kuhn Professor of Mathematics (Committee Member)
More informationThe Fibonacci hypercube
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 40 (2008), Pages 187 196 The Fibonacci hypercube Fred J. Rispoli Department of Mathematics and Computer Science Dowling College, Oakdale, NY 11769 U.S.A. Steven
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 informationMinimum Vector Rank and Complement Critical Graphs
Minimum Vector Rank and Complement Critical Graphs Xiaowei Li, Rachel Phillips September 23, 2011 Abstract The minimum rank problem is to find the smallest rank of a collection of matrices which are related
More informationarxiv: v2 [math.co] 13 Aug 2013
Orthogonality and minimality in the homology of locally finite graphs Reinhard Diestel Julian Pott arxiv:1307.0728v2 [math.co] 13 Aug 2013 August 14, 2013 Abstract Given a finite set E, a subset D E (viewed
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 informationChromatic Transversal Domatic Number of Graphs
International Mathematical Forum, 5, 010, no. 13, 639-648 Chromatic Transversal Domatic Number of Graphs L. Benedict Michael Raj 1, S. K. Ayyaswamy and I. Sahul Hamid 3 1 Department of Mathematics, St.
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 informationAlgorithm and Complexity of Disjointed Connected Dominating Set Problem on Trees
Algorithm and Complexity of Disjointed Connected Dominating Set Problem on Trees Wei Wang joint with Zishen Yang, Xianliang Liu School of Mathematics and Statistics, Xi an Jiaotong University Dec 20, 2016
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 informationSome new results on circle graphs. Guillermo Durán 1
Some new results on circle graphs Guillermo Durán 1 Departamento de Ingeniería Industrial, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile gduran@dii.uchile.cl Departamento
More informationEternal Domination: Criticality and Reachability
Eternal Domination: Criticality and Reachability William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224-2669 wkloster@unf.edu Gary MacGillivray Department of Mathematics
More informationOn Locating Domination Number of. Boolean Graph BG 2 (G)
International Mathematical Forum, Vol. 12, 2017, no. 20, 973-982 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7977 On Locating Domination Number of Boolean Graph BG 2 (G) M. Bhanumathi
More informationEdge colorings. Definition The minimum k such that G has a proper k-edge-coloring is called the edge chromatic number of G and is denoted
Edge colorings Edge coloring problems often arise when objects being scheduled are pairs of underlying elements, e.g., a pair of teams that play each other, a pair consisting of a teacher and a class,
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 informationRecognizing Interval Bigraphs by Forbidden Patterns
Recognizing Interval Bigraphs by Forbidden Patterns Arash Rafiey Simon Fraser University, Vancouver, Canada, and Indiana State University, IN, USA arashr@sfu.ca, arash.rafiey@indstate.edu Abstract Let
More informationThe Matrix-Tree Theorem and Its Applications to Complete and Complete Bipartite Graphs
The Matrix-Tree Theorem and Its Applications to Complete and Complete Bipartite Graphs Frankie Smith Nebraska Wesleyan University fsmith@nebrwesleyan.edu May 11, 2015 Abstract We will look at how to represent
More informationSANDRA SPIROFF AND CAMERON WICKHAM
A ZERO DIVISOR GRAPH DETERMINED BY EQUIVALENCE CLASSES OF ZERO DIVISORS arxiv:0801.0086v2 [math.ac] 17 Aug 2009 SANDRA SPIROFF AND CAMERON WICKHAM Abstract. We study the zero divisor graph determined by
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 informationFOUR EDGE-INDEPENDENT SPANNING TREES 1
FOUR EDGE-INDEPENDENT SPANNING TREES 1 Alexander Hoyer and Robin Thomas School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332-0160, USA ABSTRACT We prove an ear-decomposition theorem
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 informationDegree Equitable Domination Number and Independent Domination Number of a Graph
Degree Equitable Domination Number and Independent Domination Number of a Graph A.Nellai Murugan 1, G.Victor Emmanuel 2 Assoc. Prof. of Mathematics, V.O. Chidambaram College, Thuthukudi-628 008, Tamilnadu,
More informationMinimum Number of Palettes in Edge Colorings
Graphs and Combinatorics (2014) 30:619 626 DOI 10.1007/s00373-013-1298-8 ORIGINAL PAPER Minimum Number of Palettes in Edge Colorings Mirko Horňák Rafał Kalinowski Mariusz Meszka Mariusz Woźniak Received:
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 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 informationInduced Subgraph Saturated Graphs
Theory and Applications of Graphs Volume 3 Issue Article 1 016 Induced Subgraph Saturated Graphs Craig M. Tennenhouse University of New England, ctennenhouse@une.edu Follow this and additional works at:
More informationCharacterization of Super Strongly Perfect Graphs in Chordal and Strongly Chordal Graphs
ISSN 0975-3303 Mapana J Sci, 11, 4(2012), 121-131 https://doi.org/10.12725/mjs.23.10 Characterization of Super Strongly Perfect Graphs in Chordal and Strongly Chordal Graphs R Mary Jeya Jothi * and A Amutha
More informationThe Q-matrix completion problem
Electronic Journal of Linear Algebra Volume 18 Volume 18 (2009) Article 15 2009 The Q-matrix completion problem Luz Maria DeAlba, luz.dealba@drake.edu Leslie Hogben Bhaba Kumar Sarma Follow this and additional
More informationPAIRED-DOMINATION. S. Fitzpatrick. Dalhousie University, Halifax, Canada, B3H 3J5. and B. Hartnell. Saint Mary s University, Halifax, Canada, B3H 3C3
Discussiones Mathematicae Graph Theory 18 (1998 ) 63 72 PAIRED-DOMINATION S. Fitzpatrick Dalhousie University, Halifax, Canada, B3H 3J5 and B. Hartnell Saint Mary s University, Halifax, Canada, B3H 3C3
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 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 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 information