DP color functions versus chromatic polynomials (II)

Abstract

AbstractFor any connected graph , let and denote the chromatic polynomial and Dvořák and Postle (DP) color function of , respectively. It is known that holds for every positive integer . Let (resp., ) be the set of graphs for which there exists an integer such that (resp., ) holds for all integers . Determining the sets and is an important open problem on the DP color function. For any edge set of , let be the size of a shortest cycle in such that is odd if such a cycle exists, and otherwise. We denote as if . In this paper, we prove that if has a spanning tree such that is odd for each , the edges in can be labeled as with for all and each edge is contained in a cycle of size with , then is a graph in . As a direct application, all plane near‐triangulations and complete multipartite graphs with at least three partite sets belong to . We also show that if is a set of edges in such that is even and satisfies certain conditions, then belongs to . In particular, if , where is a set of edges between two disjoint vertex subsets of , then belongs to . Both results extend known ones by Dong and Yang.