The rainbow numbers of paths in maximal bipartite planar graphs

Abstract

Given two graphs G and T, the rainbow number of T in G, denoted by $rb(G,T)$, is the minimum positive integer t such that, if G contains a copy of T, then every t-edge-coloring of G contains a rainbow copy of T. Given a family of graphs $G$ and a graph T, if every graph in $G$ contains a copy of T, then the rainbow number of T in $G$, denoted by $rb(G,T)$, is defined as $\max \left\{rb\right(G,T\left)\right|G\in G\}$. Given a graph T, let $H_n\left(T\right)$ denote the family of all maximal bipartite planar graphs on n vertices that contain a copy of T. In this paper, we study the rainbow numbers of paths in maximal bipartite planar graphs, we get the exact value of $rb\left(H_n\right(P_{\ell }),P_{\ell })$ for $n\ge \ell$ and $\ell \ne 8$, and the lower bound of $rb\left(H_n\right(P_8),P_8)$ for all $n\ge 8$.