Strong Rainbow Edge coloring of Necklace Graphs

Transcription

1 Volume 109 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Strong Rainbow Edge coloring of Necklace Graphs G.Vidya 1 and Indra Rajasingh 1, 1 School of Advanced Sciences, VIT University, Chennai , India vidyaganesan15@gmail.com Abstract A rainbow edge coloring of a connected graph is a coloring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are colored the same. The minimum number of colors required to make G strongly rainbow connected is known as strong rainbow connection number and is denoted by src(g). We give algorithms for strong rainbow edge coloring of necklace graphs and glued trees using optimum number of colors. AMS Subject Classification: 05C15 Keywords: rainbow coloring; strong rainbow coloring; necklace graphs Glued trees 1 Introduction Let G be a nontrivial connected graph on which an edge-coloring c : E(G) {1, 2,... n}, n N, is defined, where adjacent edges may be colored the same. A path is a rainbow if no two edges of it are colored the same. An edge-coloring graph G is rainbow connected if any two vertices are connected by a rainbow path. An edge coloring under which G is rainbow connected is called a rainbow coloring. An edge colored graph G is strongly rainbow connected if for every pair of distinct vertices, there exists atleast one shortest rainbow path Thus, the rainbow connection number of a connected graph G, denoted by rc(g), is the smallest number of colors that are needed in order to make G rainbow connected and the strong rainbow connection number is the minimum number of colors that makes G strongly rainbow connected denoted by src(g). Thus rc(g) src(g) for every connected graph G

2 The concept of rainbow coloring was introduced by Chartand et al in 2008 [1]. The rainbow connection number is not only a natural combinatorial measure, but it also has applications to the secure transfer of classified information between agencies. The rainbow connections of graphs are very new concepts and there has been great interest in these concepts and a lot of results have been published [1]. Precise values of rainbow connection number for many special graphs like complete multipartite graphs, Peterson graph and wheel graph were also determined. It was shown in Chakraborty et al.,in [2], that computing the rainbow connection number of an arbitrary graph is NP-Hard. 2 Basic Concepts Definition 2.1. [4] Let K m and K ti be complete graphs on m and t i vertices respectively, 1 i m. The graph obtained by identifying any one vertex of K ti with the i th vertex of K m, 1 i m, is called a K-necklace and is denoted by KN(K m ; K t1, K t2,..., K tm ). Definition 2.2. [3] The windmill graph W d(n, m), n, m 2, is constructed by joining m copies of the complete graph K n at a shared vertex. See Fig. 2. For our convenience we denote the windmill graph W d(n, m) as W M(K t1, K t2,..., K tm ), where K ti is a complete graph on t i vertices, t i 2, 1 i m, m 2. Definition 2.3. [4] Let P m be a path on m vertices and K ti be a complete graph on t i vertices, 1 i m. The graph obtained by identifying any one vertex of K ti with the i th vertex of P m, 1 i m, is called a P -necklace and is denoted by P N(P m ; K t1, K t2,..., K tm ). See Fig. 3. Definition 2.4. [4] Let C m be a cycle on m vertices and let K ti be a complete graph on t i vertices, 1 i m. The graph obtained by identifying any one vertex of K ti with the i th vertex of C m, 1 i m, is called a C-necklace and is denoted by CN(C m ; K t1, K t2,..., K tm ). See Fig Main Results Proposition 3.1. [1] Let G be a nontrivial connected graph of size m. Then 2 192

3 (i) src(g) = 1 if and only if G is a complete graph; (ii) rc(g) = src(g) = m if and only if G is a tree; (iii) rc(c n ) = src(c n ) = n 2 for each integer n 4, where Cn is a cycle with size n. Theorem 3.2. Let G be the K-necklace KN(K m ; K t1, K t2,..., K tm ), m 1, t i 4, 1 i m. Then src(g) = v + 1. Proof. Let v 1, v 2,..., v m be the vertices of the subgraph K m of G such that each complete graph K ti, shares exactly one vertex v i of K m, 1 i m. By Proposition 3.1, If G is a complete graph then src(g) = 1. Consider the vertices u, v and w in G. Let us assume that one vertex u is in any one of the copies of K ti say K t1. Let v be in another copy of K ti say K t2. Let w be a vertex in K m.the uwv is a geodesic rainbow path between u and v in G. This implies that distinct copies of K ti and subgraph K m of G receives distinct colors. Therefore src(g) = V (K m ) + 1. See Fig.1. K t1 v 1 K t4 K 4 v 2 v 4 K t2 v 3 K t3 Figure 1: KN(K 4 ; K t1, K t2, K t3, K t4 ) Theorem 3.3. Let G be a windmill graph W M(K t1, K t2,..., K tm ), m 2, t i 4, 1 i m. Then src(g) = K ti. Proof. Let w be the cut-vertex of G. The windmill graph is constructed by joining m copies of complete graph to the shared vertex. Consider two vertices u and v in G. Let us assume that one vertex u is in any 3 193

4 one of the copies of K ti say K t1. Let v be in another copy of K ti say K t2. The uwv is a geodesic rainbow path between u and v. This implies that distinct copies of K ti receive distinct colors, 1 i m. Thus src(g) = Kti. See Fig.2. K t1 w K t4 K t2 K t3 Figure 2: W M(K t1, K t2, K t3, K t4 ) Theorem 3.4. Let G be the P -necklace graph P N(P m ; K t1, K t2,..., K tm ), m 3, t i 4, 1 i m. Then src(g) = 2 V (P m ) 1. Proof. Let v 1, v 2,..., v m be the vertices of the subgraph P m of G such that each complete graph K ti shares exactly one vertex v i of P m, 1 i m. Consider the vertices u and v and w in G. Let us assume that one vertex u is in any one of the copies of K ti say K t1. Let v be in another copy of K ti say K t2. Let w be a vertex in P m.the uwv is a geodesic rainbow path between u and vin G. This implies that distinct copies of K ti and the path P m receives distinct colors, 1 i m. Therefore src(g) = 2 V (P m ) 1. See Fig.3. Theorem 3.5. Let G be a C-necklace graph CN(C m ; K t1, K t2,..., K tm ), m 3, t i 4, 1 i m on n(= m i=1 t i ) vertices. Then src(g) = src(c n ) + V (C n ). Proof. Let v 1, v 2,..., v m be the vertices of the subgraph C m of G such that each complete graph K ti, shares exactly one vertex v i of C m, 1 i m. By Proposition 3.1, If C n is a cycle on n vertices, n 4 then src(g) = n 2. Consider the vertices u, v and w in G. Let us assume that one vertex u i s in any one of the copies of K ti say K t1. Let v be in another copy of K ti say K t2. Let w be a vertex in C n The 4 194

5 uwv is a geodesic rainbow path between u and v in G. This implies that distinct copies of K ti receive distinct colors. Therefore src(g) = src(c n ) + V (C n ). See Fig.4. K K K K K t 1 t 2 t 5 t 8 t 9 v 1 v 2 v 5 v 8 v 9 Figure 3: P N(P 9 ; K t1,..., K t9 ) K t1 v 1 K t m v m C m v 2 K t2 v 3 K t3 Figure 4: CN(C m ; K t1, K t2,..., K tm ) 4 Glued Trees Glued binary trees were introduced by physicists as a tool to design quantum algorithms [5] and quantum circuits [19]. It plays a significant 5 195

6 role in Quantum Information Theory [7]. It is also used to study the transmission properties of continuous time quantum walks in quantum physics [5, 8]. Lockhart et al. [9] designed glued tree algorithm using glued binary trees. An r level complete binary tree T (r) has 2 r leaves. An r-level glued binary tree GT (r) is formed by connecting the leaves of two r-level complete binary trees T 1 (r) and T 2 (r). In general, the vertex set of GT (r) is the union of the vertex sets of complete binary trees T 1 (r) and T 2 (r). Let L 1 and L 2 denote the vertex sets of leaves of T 1 (r) and T 2 (r) respectively. The sets L 1 and L 2 induce a bipartite graph in GT(r). Feder [7] classifies glued binary trees into those without randomization and those with randomization. If the edges of the bipartite subgraph induced by the sets L 1 and L 2 are in some fixed order, then it is called a glued binary trees without randomization, while if the sets L 1 and L 2 induce an arbitrary bipartite graph, then it is called a glued binary trees with randomization. In what follows, we consider a glued trees without randomization, where the subgraphs induced by L 1 and L 2 are 2 r 1 disjoint 4-cycles. Theorem 4.1. Let G be a binary glued tree. Then src(g) = m. Proof. By Proposition 3.1, If G is a tree of size m then src(g) = m. The edges of the disjoint 4-cycles form an edge cut of G, whose removal yield two components, each isomorphic to a binary tree, say T 1 (r) and T 2 (r). Each of E(T 1 ) and E(T 2 ) are colored with E(T 1 ) distinct colors. Consider a 4-cycle C 1 induced by vertices of L 1 and L 2. Let u and v be vertices of L 1 in C 1. Let w be the parent of u and v in L 1. Further let p be the parent of w and q be the other child of p. Label the parallel edges of C 1 incident at u and v as l(wv) and l(wu). Label the other two edges of C 1 as l(pq) and l(pw). Repeat this procedure for every 4-cycle induced by L 1 and L 2. See Fig

7 1 2 p w q u v 4 C Figure 5: Strong Rainbow Edge Coloring of Glued Trees 5 Conclusion In this paper the strong rainbow connection number of necklace graphs and glued trees have been found. Strong rainbow connection number of certain interconnection networks is under investigation. References [1] G.Chartrand, G.L. Johns, K.A. Mckeon and P.Zhang, Rainbow connection in graphs, Math. Bohemica 133, (2008), [2] S.Chakraborty, E.Fischer, A.Matsliah and R.Yuster, Hardness and algorithms for rainbow connection, Journal of Combinatorics Optimization, (2009),1-18. [3] Gallian, J.A., A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics, 17, (2014), [4] R. Jayagopal, R. Sundara Rajan and Indra Rajasingh, Tight Lower Bound For Locating-total Domination Number, International Journal of Pure and Applied Mathematics, 101, (2015),

8 [5] A.M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, D.A. Spielman, Exponential algorithmic speedup by quantum walk, Proc, 35th ACM Symposium on Theory of Computing, (STOC 2003), [6] B.L. Douglas, J.B.Wang, Efficient quantum circuit implementation of quantum walks, Phys. Rev. A, 79, (2009), [7] D. Feder, Graphs in Quantum Information Theory, Manuscript, University of Calgary, Alberta, Canada, (2015). [8] Z-J. Li, J.B. Wang, An analytical study of quantum walk through glued-tree graphs, J. Phys. A, 48, (2015), [9] J. Lockhart, C. Di Franco and M. Paternostro, Glued trees algorithm under phase damping, Physics Lett, A, 378, (2014),

9 199

10 200