International Journal of Pure and Applied Mathematics Volume 101 No. 6 2015, 873-881 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP ijpam.eu RAINBOW COLORING OF SHADOW GRAPHS Annammal Arputhamarya1, M. Helda Mercy2 1Sathyabama University Chennai, 600119, INDIA 2Panimalar Engineering College Chennai, 600123, INDIA Abstract: A rainbow 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. Computing the rainbow connection number of a graph is NP-hard and it finds its applications to the secure transfer of classified information between agencies, cellular networks, chemical compounds and scheduling. In this paper rainbow coloring of shadow graph of path graph D2(Pn) and m-shadow graph of path graph Dm(Pn) are considered and rc(G) and rvc(G) of these graphs are computed. AMS Subject Classification: 05C15 Key Words: rainbow connection number, shadow graph of path graph 1. Introduction The rainbow connection number can be motivated by its interesting interpreta- tion in the area of networking. This new concept comes from the communication of information between agencies of government. The Department of Homeland Security of USA was created in 2003 in response to weakness in the transfer of information in September 2011 terrorist attacks. Those deadly attacks took place because of the law enforcement and intelligence agencies could not com- municate with each other through their regular channels, from radio systems to databases. Also the officers and agents were unable to cross check infor- mation between various organizations. Rainbow connection has an interesting c 2015 Academic Publications, Ltd. Received: March 12, 2015 url: www.acadpubl.eu 874 A. Arputhamarya, M.H. Mercy application for the secure transfer of classified information between agencies. Consider a network G(e.g., a cellular network). To route messages between any two vertices in a pipeline, assign a distinct channel to each link(e.g., a distinct frequency). We need to minimize the number of distinct channels that we use in the network. The minimum number of distinct channels is called the rainbow connection number and is denoted by rc(G). There are also the concepts of strong rainbow connection or rainbow diameter, the rainbow connectivity, and the rainbow index. ll the graphs considered in this paper are finite, undirected and simple. 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 rainbow if no two edges of it are colored the same. In a vertex colored graph a path is said to be a rainbow path if its internal vertices have distinct colors. An edge-colored graph G is rainbow connected if any two distinct vertices are connected by at least one rainbow path. An edge-coloring under which G is rainbow connected is called a rainbow coloring. Clearly, if a graph is rainbow connected, it must be connected. Conversely, every connected graph has a trivial edge-coloring that makes it rainbow con- nected by coloring edges with distinct colors. 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. A rainbow col- oring using rc(G) colors is called a minimum rainbow coloring. The rainbow connection number can be viewed as a new kind of chromatic index. A vertex colored graph G is rainbow vertex connected if any two vertices are connected by atleast one rainbow path. The rainbow vertex connection number of a connected graph G denoted by rvc(G) is the smallest number of colors that are needed in order to make G rainbow vertex connected. A vertex colored graph G is strongly rainbow vertex connected if for every pair of distinct vertices, there exists a shortest rainbow path. 2. Overview Of the Paper The concept of rainbow coloring was introduced by Chartrand et al.,[4]. 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.,[3] that computing the rainbow connection RAINBOW COLORING OF SHADOW GRAPHS 875 number of an arbitrary graph is NP-Hard. To rainbow color a graph it is enough to ensure that every edge of some spanning tree in the graph gets a distinct color. Hence order of the graph minus one is an upper bound for rainbow connection number[6]. Similar to the concept of rainbow connection number, Krivelevich and Yuster[5] proposed the concept of rainbow vertex connection. Note that rvc(G) ≥ diam(G) − 1 and with equality if the diameter is 1 or 2. Krivelevich and Yuster [5] also proved that if G is a graph with n vertices and minimum degree δ, then rvc(G) < 11n/δ. In this paper rainbow coloring of shadow graph of path graph D2(Pn) and m-shadow graph of path graph Dm(Pn) are considered and rc(G) and rvc(G) of these graphs are computed. 3. Rainbow coloring of shadow graph of path graph D2(Pn): 3.1. Shadow Graph [8] The shadow graph D2(G) of a connected graph G is constructed by taking two copies of G say G′ and G′′. Join each vertex u′ in G′ to the neighbors of the ′ ′′ corresponding vertex v in G . The shadow graph D2(Pn) is constructed by ′ ′′ taking 2 copies of Pn say G and G . Let u1, u2, u3, ..., un be the vertices of first ′ copy of Pn say G and v1, v2, v3, ..., vn be the vertices of the second copy of Pn ′′ ′ i.e G . Then |V (G)| = 2n and |E(G)| = 4(n − 1). Join each vertex ui in G ′′ to the neighbour of the corresponding vertex vi in G . Figure 1 shows shadow graph of path graph. Figure 1: D2(P3) Theorem 1. The shadow graph of a path graph admits a rainbow coloring and (i) Rainbow connection number rc(D2(Pn)) = n = src(D2(Dn)) for n ≥ 2 876 A. Arputhamarya, M.H. Mercy (ii) Rainbow vertex connection number rvc(D2(Pn)) = 1 = srvc(D2(Pn)) for n=2 and for n ≥ 3, rvc(D2(Pn)) = n − 2 = srvc(D2(Pn)). Proof. If n = 2, it is obvious that rc(D2(Pn)) = 2 = src(D2(Pn)). Next we shall prove that the rainbow connection number for n ≥ 3 is n. That is rc(D2(Pn)) = n = src(D2(Pn)). For convenience define the graph D2(Pn) by G. For a path graph Pn , diameter = n − 1. We know that for any graph G the rainbow connection number rc(G) ≥ diam(G) [5], therefore rc(G) ≥ n − 1. Edge Coloring Algorithm: 1. Color the upper spine (u1, u2), (u2, u3), ..., (un−1, un), edges of G with col- ors c1, c2, ..., cn−1 respectively. 2. Color the lower spine (v1, v2), (v2, v3), ..., (vn−1, vn), edges of G with colors c1, c2, ..., cn−1 respectively. 3. Color the edges (ui, vi+1) and (ui+1, vi) where 1 ≤ i ≤ n − 1 with the color cn. Consider any distinct path in D2(Pn): Case(i): If P1 is any path connecting ui and uj, 1 ≤ i, j ≤ n , i 6= j, then the shortest path connecting the vertices ui and uj form a rainbow path. Case(ii): Consider a path P2 connecting vi and vj, 1 ≤ i, j ≤ n , i 6= j, then the shortest path connecting the vertices vi and vj form a rainbow path. Case(iii): Consider a path P3 connecting ui and vj, 1 ≤ i, j ≤ n , i 6= j, then the path connecting such vertices form a rainbow path.. Thus if we consider any path connecting the vertices there exist at least one path in G that forms a rainbow path. Therefore if we color the edges of G, the rainbow connection number rc(G) = n. Also the shortest path connecting any two vertices forms a rainbow path, hence the rainbow coloring is a strong rainbow coloring. Therefore rc(G) = src(G) = n. We see that in Figure 2 , rc(G) = 6 = src(G). Now we shall prove for the rainbow vertex connection number. Rainbow vertex coloring is similar to the concept of rainbow edge coloring. In rainbow edge coloring the edges are assigned distinct colors such that the path connect- ing any two vertices form a rainbow path. Here we have to prove that every two vertices are connected by a path whose internal vertices have distinct colors and hence forming a rainbow path. If G is a graph of order n, rvc(G) ≤ n − 2. Also rvc(G) ≥ diam(G) − 1[5]. For a path graph Pn, diameter = n − 1. Therefore rvc(G) ≥ n − 2. For n=2 , it is easy to see that rc(D2(Pn)) = 1 = src(D2(Pn)). Next we shall prove for n ≥ 3. RAINBOW COLORING OF SHADOW GRAPHS 877 Figure 2: D2(P6) Vertex Coloring Algorithm: 1. Color the end vertices u1, un, v1, vn with c1. 2. Color the interval values u2, u3, ..., un−1 with c1, c2, ..., cn−2 respectively and v2, v3, ..., vn−1 with c1, c2, ..., cn−2 respectively. Above said coloring is a rainbow vertex coloring and rvc(G) = n − 2. This coloring will also be a strong rainbow vertex coloring. The bound in the above theorem is sharp because rvc(G) ≥ n − 2. Figure 3: D2(P7) In Figure 3, rvc(D2(P7)) = 5 = srvc(D2(P7)).