In the article, the existence of rainbow cycles in edge colored plane triangulations is studied. It is shown that the minimum number rb(Tn,C3) of colors that force the existence of a rainbow C3 in any n-vertex plane triangulation is equal to 3n−4 2 . For k ≥ 4 a lower bound and for k ∈ {4,5} an upper bound of the number rb(Tn,Ck ) is determined. C © 2014 Wiley Periodicals, Inc. J. Graph Theory ...

An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. A graph G is called rainbow k-connected, if there is an edge-colouring of G with k colours such that G is rainbow-connected. In this talk we will study rainbow k-connected graphs with a minimum number of edges. For an integer n ≥ 3 and 1 ≤ k ≤ n− 1 let t(n, k) denote the ...

Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by γc(G) + 2, where γc(G) is th...

An edge-coloured path is rainbow if all of its edges have distinct colours. For a connected graph G, the rainbow connection number rc(G) of G is the minimum number of colours in an edge-colouring of G such that, any two vertices are connected by a rainbow path. Similarly, the strong rainbow connection number src(G) ofG is the minimum number of colours in an edge-colouring of G such that, any tw...

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). G...

For the terminology and notations not defined here, we adopt those in Bondy and Murty [1] and Xu [2] and consider simple graphs only. Let G = (V,E) be a graph with vertex set V = V (G) and edge set E = E(G). For any vertex v ∈ V , NG(v) denotes the open neighborhood of v in G and NG[v] = NG(v) ∪ {v} the closed one. dG(v) = |NG(v)| is called the degree of v in G, ∆ and δ denote the maximum degre...