The 3-rainbow index of graph operations

A tree T , in an edge-colored graph G, is called a rainbow tree if no two edges of T are assigned the same color. A k-rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S ⊆ V (T ). The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G, denoted by rxk(G). Graph operations, both binary and unary, are an interesting subject, which can be used to understand structures of graphs. In this paper, we will study the 3-rainbow index with respect to three important graph product operations (namely Cartesian product, strong product, lexicographic product) and other graph operations. Firstly, let Gi(i = 1, 2, · · · , k) be connected graphs and G∗ be the Cartesian product of Gi. That is to say, G∗ = G1 G2 · · · Gk (k ≥ 2). Then we proved that rx3(G ∗) ≤ ∑k i=1 rx3(Gi). And we also get the condition when the equality holds. As a corollary, we obtain an upper bound for the 3-rainbow index of strong product graph. Secondly, we discuss the 3-rainbow index of the lexicographic graph G[H] for connected graphs G and H . And the sharp upper bound is given. Finally, we consider some other simple graph operations : the join of two graphs, split a vertex of a graph and subdivide an edge of a graph. The upper bounds of the 3-rainbow index of the three operation graphs are presented, respectively. Key–Words: 3-rainbow index; Cartesian product; strong product; lexicographic product.