Planar anti-Ramsey numbers for paths and cycles (original) (raw)

2017, Cornell University - arXiv

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 T n (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 of H, denoted ar P (n, H), is the maximum number of colors in an edge-coloring of a plane triangulation T ∈ T n (H) such that T contains no rainbow copy of H. Analogous to anti-Ramsey numbers and Turán numbers, planar anti-Ramsey numbers are closely related to planar Turán numbers, where the planar Turán number of H is the maximum number of edges of a planar graph on n vertices without containing H as a subgraph. The study of ar P (n, H) (under the name of rainbow numbers) was initiated by Horňák, Jendrol ′ , Schiermeyer and Soták [J Graph Theory 78 (2015) 248-257]. In this paper we study planar anti-Ramsey numbers for paths and cycles. We first establish lower bounds for ar P (n, P k) when n ≥ k ≥ 8. We then improve the existing lower bound for ar P (n, C k) when k ≥ 5 and n ≥ k 2 − k. Finally, using the main ideas in the above-mentioned paper, we obtain upper bounds for ar P (n, C 6) when n ≥ 8 and ar P (n, C 7) when n ≥ 13, respectively.