Rainbow numbers for small graphs in planar graphs

Abstract 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 r b ( 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 T n 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 T n . Furthermore, we give upper bounds of rainbow numbers for triangles with two pendant edges with respect to T n .