Rainbow numbers for small graphs in planar graphs

摘要

Let G be a family of graphs and H be a subgraph of at least one of the graphs in G. The rainbow number for H with respect to G, denoted rb(G,H), is the minimum number k such that, if H⊆G∈G, then any k-edge-coloring of G contains a rainbow H (i.e., any two edges of H are colored distinct). Denote by Tn the class of all plane triangulations of order n and Wd the wheel graph of order d+1. In this paper, we determine the exact rainbow numbers for matchings and a triangle with one or two pendant edges with respect to Wd, and the exact rainbow numbers for the triangle with one pendant edge with respect to Tn. Furthermore, we give upper bounds of rainbow numbers for triangles with two pendant edges with respect to Tn.