Chapter 4. Triangular Sum Labeling

Transcription

1 Chapter 4 Triangular Sum Labeling 32

2 Chapter 4. Triangular Sum Graphs Introduction This chapter is focused on triangular sum labeling of graphs. As every graph is not a triangular sum graph it is very interesting to investigate graphs or graph families which are not triangular sum graphs. Here we investigate some of the graphs which are not triangular sum graphs. 4.2 Triangular Sum Graphs Definition A triangular number is a number obtained by adding all positive integers less than or equal to a given positive integer n. If n th triangular number is denoted by T n then T n = 1 n(n + 1). It is easy to observe that there does not exist consecutive 2 integers which are triangular numbers. The first few triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55. Definition A function f is called a triangular sum labeling of a graph G if f : V (G) N ( where N is the set of all non-negative integers) is injective and the induced function f + : E(G) {T 1,T 2,...,T p } defined as f + (uv) = f (u)+ f (v) for every edge e having end vertices is bijective. A graph which admits triangular sum labeling is called a triangular sum graph. The notion of triangular sum labeling was originated by Hegde and Shankaran [27]. Illustration A triangular sum labeling of P 7 is shown in the Figure 4.1. FIGURE 4.1: P 7 and its triangular sum labeling

3 Chapter 4. Triangular Sum Graphs Some Existing Results on Triangular Sum Graphs Hegde and Shankaran [27] have proved that path P n is a triangular sum graph. star K 1,n is a triangular sum graph. any tree obtained from the star K 1,n by replacing each edge by a path is a triangular sum graph. the lobster T obtained by joining the centers of k copies of a stat to a new vertex w is a triangular sum graph. the complete n-ary tree T m of level m is a triangular sum graph. the complete graph K n is triangular sum if and only if n 2. if G is an Eulerian (p,q)-graph admitting a triangular sum labeling then q 1(mod 12). the Dutch windmill DW(n) ( n copies of K 3 sharing a common vertex) is not a triangular sum graph. the complete graph K 4 can be embedded as an induced subgraph of a triangular sum graph. if G is a unicyclic graph consisting of a unique triangle (v 1,v 2,v 3,v 1 ) with deg(v 2 ) = deg(v 2 ) = 3, a path P = v 1,u 1,u 2,...,u n of length n and k pendant vertices w 1,w 2,...,w k adjacent to v 1 then G is a triangular sum graph for all n 3. Hegde and Shankaran [27] have conjectured that The complete graph K n, n 5 is a forbidden subgraph for a triangular sum graph. Vaidya et al [71] have proved that every cycle can be embedded as an induced subgraph of a triangular sum graph. every cycle with one chord can be embedded as an induced subgraph of a triangular sum graph.

4 Chapter 4. Triangular Sum Graphs 35 every cycle with twin chords can be embedded as an induced subgraph of a triangular sum graph. Seoud and Salim [49] have proved that every tree can be embedded as a triangular sum graph. P n1 P n2, for any n 1 > 4 and for any n 2 is a triangular sum graph. P n, for n 5 is a triangular sum graph. the graph which is constructed by attaching the roots of different stars to one vertex, is a triangular sum graph. the tree which is obtained by identifying the roots of n stars: K 1,t1, K 1,,t2,..., K 1,tn with the n vertices of a path P n, is a triangular sum graph. the graph P n K m is a triangular sum graph. every symmetrical tree is a triangular sum graph. all the trees of order 9 are triangular sum graphs. the arbitrary union of C 4 can be embedded as an induced subgraph of a triangular sum graph. the one-point union of m cycles of length 4 can be embedded as an induced subgraph of a triangular sum graph. Seoud and Salim [49] have shown a construction to generate many triangular sum trees from a triangular sum tree of order p by joining vertices to every vertex of a label less than p. Murugesan et al [44] have defined second and third order triangular sum labeling. Murugesan et al [44] have proved that the path P n admits second order triangular sum labeling. the star graph K 1,n admits second order triangular sum labeling. coconut trees admit second order triangular sum labeling. the bistar B m,n admits second order triangular sum labeling.

5 Chapter 4. Triangular Sum Graphs 36 the graph B m,n,k (a graph obtained from a path of length k by attaching the stars K 1,m and K 1,n with its pendent vertices) admits second order triangular sum labeling. the path P n admits third order triangular sum labeling. the star graph K 1,n admits third order triangular sum labeling. coconut trees admit third order triangular sum labeling. the bistar B m,n admits third order triangular sum labeling. the graph B m,n,k admits third order triangular sum labeling 4.4 Some New Results on Triangular Sum Labeling In the present work we investigate some classes of graphs which does not admit a triangular sum labeling. Lemma In every triangular sum graph G the vertices with labels 0 and 1 are always adjacent. Proof. The edge label T 1 = 1 is possible only when the vertices with label 0 and 1 are adjacent. Lemma In any triangular sum graph G, 0 and 1 cannot be the labels of vertices of the same triangle contained in G. Proof. Let a 0,a 1, and a 2 be the vertices of a triangle. If a 0 and a 1 are labeled with 0 and 1 respectively and a 2 is labeled with some x N, where x 0, x 1. Such vertex labeling will give rise to edge labels with 1,x, and x + 1. In order to admit a triangular sum labeling, x and x + 1 must be triangular numbers. But it is not possible as we mentioned in Definition Lemma In any triangular sum graph G, 1 and 2 cannot be the labels of vertices of the same triangle contained in G.

6 Chapter 4. Triangular Sum Graphs 37 Proof. Let a 0,a 1,a 2 be the vertices of a triangle. Let a 0 and a 1 are labeled with 1 and 2 respectively and a 2 is labeled with some x N, where x 1,x 2. Such vertex labeling will give rise to edge labels 3,x + 1, and x + 2. In order to admit a triangular sum labeling, x + 1 and x + 2 must be triangular numbers. Which is not possible due to the fact mentioned in Definition Theorem The Helm graph H n is not a triangular sum graphs. Proof. Let us denote the apex vertex as c 1, the consecutive vertices adjacent to c 1 as v 1,v 2,...,v n, and the pendant vertices adjacent to v 1,v 2,...,v n as u 1,u 2,...,u n respectively. If possible H n admits a triangular sum labeling f : V N, then we consider following cases: Case 1: f (c 1 ) = 0. Then according to Lemma 4.4.1, we have to assign label 1 to exactly one of the vertex from v 1,v 2,...,v n. Then there is a triangle having the vertices with labels 0 and 1 as adjacent vertices, which contradicts the Lemma Case 2: Any one of the vertex from v 1,v 2,...,v n is labeled with 0. Without loss of generality let us assume that f (v 1 ) = 0. Then one of the vertex from c 1,v 2,v n,u 1 must be labeled with 1. Note that each of the vertex from c 1,v 2,v n,u 1 is adjacent to v 1. Subcase 1: If one of the vertex from c 1,v 2,v n is labeled with 1. In each of the possibilities, there is a triangle having two of the vertices with labels 0 and 1, which contradicts the Lemma Subcase 2: If f (u 1 ) = 1 then the edge label T 2 = 3 can be obtained by vertex labels 0, 3 or 1, 2. The vertex with label 1 and the vertex with label 2 cannot be adjacent as u 1 is a pendant vertex having label 1 and it is adjacent to the vertex with label 0. Therefore one of the vertex from v 2,v n,c 1 must receive the label 3. Thus there is a triangle whose two of the vertices are labeled with 0 and 3. Let the third vertex be labeled with x, with x 0 and x 3. To admit a triangular sum labeling 3,x,x + 3 must be distinct triangular numbers. i.e. x and x + 3 are two distinct triangular numbers other than 3 having difference 3, which is not possible.

7 Chapter 4. Triangular Sum Graphs 38 Case 3: Any one of the vertex from u 1,u 2,...,u n is labeled with 0. Without loss of generality we may assume that f (u 1 ) = 0. Then according to Lemma 4.4.1, f (v 1 ) = 1. The edge labels T 2 = 3 can be obtained by vertex labels 0, 3 or 1, 2. The vertex with label 0 and the vertex with label 3 cannot be the adjacent vertices as u 1 is a pendant vertex having label 0 and it is adjacent to the vertex with label 1. Therefore one of the vertices from v 2,v n,c 1 must be labeled with 2. Thus we have a triangle having the vertices with labels 1 and 2 which contradicts the Lemma Thus in each of the possibilities discussed above H n does not admits a triangular sum labeling. Theorem If every edge of a graph G is an edge of a triangle then G is not a triangular sum graph. Proof. If G admits a triangular sum labeling then according to Lemma there exists two adjacent vertices having labels 0 and 1 respectively. So there is a triangle having two of the vertices labeled with 0 and 1, which contradicts Lemma Thus G does not admit a triangular sum labeling. Following are the immediate corollaries of the previous result. Corollary The wheel graph W n is not a triangular sum graph. Corollary The fan graph f n = P n 1 + K 1 is not a triangular sum graph. Corollary The friendship graph F n = nk 3 is not a triangular sum graph. Corollary The graph g n (the graph obtained by joining all the vertices of P n to two additional vertices) is not a triangular sum graph. Corollary The flower graph (the graph obtained by joining all the pendant vertices of helm graph H n with the apex vertex) is not a triangular sum graph. Corollary The graph obtained by joining apex vertices of two wheels and respective apex vertices with a new vertex is not a triangular sum graph. Theorem The graph < W n : W m > is not a triangular sum graphs.

8 Chapter 4. Triangular Sum Graphs 39 Proof. Let G =< W n : W m >. Let us denote the apex vertex of W n by u 0 and the vertices adjacent to u 0 of the wheel W n by u 1,u 2,...,u n. Similarly denote the apex vertex of other wheel W m by v 0 and the vertices adjacent to v 0 of the wheel W m by v 1,v 2,...,v m. Let w be the new vertex adjacent to apex vertices of both the wheels. If possible let f :V N be one of the possible triangular sum labeling. According to Lemma 4.4.1, 0 and 1 are the labels of some two adjacent vertices of the graph G, we have the following cases: Case 1: If 0 and 1 be the labels of adjacent vertices in W n or W m, then there is a triangle having two of the vertices labeled with 0 and 1. Which contradicts the Lemma Case 2: If f (w) = 0 then according to Lemma one of the vertex from u 0 and v 0 is labeled with 1. Without loss of generality we may assume that f (u 0 ) = 1. To have an edge label T 2 = 3 we have the following possibilities: Subcase 2.1: One of the vertices from u 1,u 2,...,u n is labeled with 2. Without loss of generality assume that f (u i ) = 2, for some i {1,2,3,...,n}. In this situation we will get a triangle having two of its vertices are labeled with 1 and 2, which contradicts the Lemma Subcase 2.2: Assume that f (v 0 ) = 3. Now to get the edge label T 3 = 6 we have the following subcases: Subcase 2.2.1: Assume that f (u i ) = 5, for some i {1,2,3,...,n}. In this situation we will get a triangle with distinct vertex labels 1,5 and x. Then x + 5 and x + 1 will be the edge labels of two edges with difference 4. It is possible only if x = 5, but x 5 as we have f (u i ) = 5. Subcase 2.2.2: Assume that 2 and 4 are the labels of two adjacent vertices from one of the two wheels. So there exists a triangle whose vertex labels are either 1,2, and 4 or 3,2, and 4. In either of the situation will give rise to an edge label 5 which is not a triangular number. Case 3: If f (w) = 1 then one of the vertices from u 0 and v 0 is labeled with 0. Without loss of generality assume that f (u 0 ) = 0. To have an edge label 3 we have the following possibilities:

9 Chapter 4. Triangular Sum Graphs 40 Subcase 3.1: If f (u i ) = 3 for some i {1,2,3,...,n}. Then there is a triangle having vertex labels as 0,3,x, with x 3. Thus we have two edge labels x + 3 and x which are two distinct triangular numbers having difference 3. So x = 3, which is not possible as x 3. Subcase 3.2: Assume that f (v 0 ) = 2. Now to obtain the edge label T 3 = 6 we have to consider the following possibilities: (i) 6=6+0; (ii) 6=5+1; (iii) 6=4+2. (i) If 6 = then one of the vertices from u 1,u 2,...,u n must be labeled with 6. Without loss of generality we may assume that f (u i ) = 6 for some i {1,2,3,...,n}. In this situation there are two distinct triangles having vertex labels 0,6,x and 0,6,y, for two distinct triangular numbers x and y each of which are different from 0 and 6. Then x+6 and x are two distinct triangular numbers having difference 6. This is possible only for x = 15. On the other hand y + 6 and y are two distinct triangular numbers having difference 6. Then y = 15. ( The x = y = 15 which is not possible as f is one-one) (ii) If 6 = and f (w) = 1, then in this situation label of one of the vertex adjacent to w must be 5. This is not possible as w adjacent to the vertices whose labels are 0 and 2. (iii) If 6 = In this case one of the vertices from v 1,v 2,...,v m is labeled with 4. Assume that f (v i ) = 4, for some i {1,2,3,...,m}. In this situation there is a triangle having vertex labels 2,4 and x (where x is a positive integer with x 2,x 4.) Then 4 + x and 2 + x will be the edge labels of two edges i.e. 4 + x and 2 + x are two distinct triangular numbers with difference 2 which is not possible. Thus we conclude that in each of the possibilities discussed above the graph G under consideration does not admit a triangular sum labeling.

10 Chapter 4. Triangular Sum Graphs Scope of Further Research To investigate triangular sum labeling for the graph obtained from some graph operations on a given graph. To investigate forbidden subgraphs characterization for the triangular sum labeling. To investigate graphs or graph families which admit triangular sum labeling. 4.6 Concluding Remarks We have investigated some graphs which are not triangular sum graphs. This work is a nice combination of graph theory and combinatorial number theory. The next chapter is aimed to discuss prime labeling of graphs.