A Mathematical Model for the Rainbow Vertex-Connection Number Fidan Nuriyeva 1 , Onur Ugurlu 2 and Hakan Kutucu 3 1 Institute of Cybernetics, Azerbaijan Academy of Sciences Baku, Azerbaijan nuriyevafidan@gmail.com 2 Department of Mathematics, Ege University Izmir, Turkey onur__ugurlu@hotmail.com 3 Department of Computer Engineering, Karabuk University Karabuk, Turkey hakankutucu@karabuk.edu.tr Abstract The concept of rainbow connection was introduced by Chartrand et al. [2] in 2008. A vertex-colored graph is rainbow vertex- connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex- connection of a connected graph G, denoted by rvc(G), is the minimum number of colors needed to make G rainbow vertex- connected. In this paper, we introduce the first mathematical model of the problem. Keywords: Rainbow Vertex Connection, Graph Coloring, Integer Programming, Mathematical Modeling. 1. Introduction An edge coloring of a graph is a function from its edge set to the set of natural numbers. A path in an edge colored graph with no two edges sharing the same color is called a rainbow path. An edge colored graph is said to be rainbow connected if every pair of vertices is connected by at least one rainbow path. Such a coloring is called a rainbow coloring of the graph. The minimum number of colors required to rainbow color a connected graph G is called its rainbow connection number, denoted by rc(G). For a basic introduction to the topic, see Chapter 11 in [3]. The concept of rainbow connectivity was recently introduced by Chartrand et al. in [2] as a measure of strengthening connectivity. The rainbow connection problem, apart from being an interesting combinatorial property, also finds an application in routing messages [4]. In addition to being a natural combinatorial measure, rainbow connectivity can be motivated by its interesting interpretation in the area of networking. Suppose that G represents a network (e.g., a cellular network). We wish to route messages between any two vertices in a pipeline, and require that each link on the route between the vertices (namely, each edge on the path) is assigned a distinct channel (e.g. a distinct frequency). Clearly, we want to minimize the number of distinct channels that we use in our network. This number is precisely rc(G) [11]. As one can see, the above rainbow connection number involves edge colorings of graphs. A natural idea is to generalize it to a concept that involves vertex-colorings. In [8], Krivelevich and Yuster are the first to introduce a new parameter corresponding to the rainbow connection number which is defined on a vertex-colored graph. A vertex-colored graph G is rainbow vertex-connected if its every two distinct vertices are connected by a path whose internal vertices have distinct colors. A vertex-coloring under which G is rainbow vertex-connected is called a rainbow vertex-coloring. The rainbow vertex-connection number of a connected graph G, denoted by rcv(G) is the smallest number of colors that are needed in order to make G rainbow vertex-connected. The minimum rainbow vertex-coloring is defined similarly [13]. Rainbow connectivity from a computational point of view was first studied by Caro et al. [4] who conjectured that computing the rainbow connection number of a given graph is NP-Hard. This conjecture was confirmed by Chakraborty et al. [11], who proved that even deciding whether rainbow connection number of a graph equals 2 is NP-Complete. They also conjectured that for every ACSIJ Advances in Computer Science: an International Journal, Vol. 2, Issue 4, No.5 , September 2013 ISSN : 2322-5157 www.ACSIJ.org 103 Copyright (c) 2013 Advances in Computer Science: an International Journal. All Rights Reserved.