Planar anti-Ramsey numbers of paths and cycles

Abstract

Motivated by anti-Ramsey numbers introduced by Erdős, Simonovits and Sós in 1975, we study the anti-Ramsey problem when host graphs are plane triangulations. Given a positive integer n and a planar graph H, let Tn(H) be the family of all plane triangulations T on n vertices such that T contains a subgraph isomorphic to H. The planar anti-Ramsey number ofH, denoted arP(n,H), is the maximum number of colors in an edge-coloring of a plane triangulation T∈Tn(H) such that T contains no rainbow copy of H. Analogously to anti-Ramsey numbers and Turán numbers, planar anti-Ramsey numbers are closely related to planar Turán numbers, where the planar Turán number ofH is the maximum number of edges of a planar graph on n vertices without containing H as a subgraph. The study of arP(n,H) (under the name of rainbow numbers) was initiated by Horňák et al. (2015). In this paper we study planar anti-Ramsey numbers for paths and cycles. We first improve existing lower bound for arP(n,Ck) when k≥5 and n≥k2−k. Then, using the main ideas in the above-mentioned paper, we obtain upper bounds for arP(n,C6) when n≥8 and arP(n,C7) when n≥13, respectively. Finally, we establish lower bounds for arP(n,Pk) when n≥k≥8.