DP color functions versus chromatic polynomials

For any graph G , the chromatic polynomial of is function P ( m ) which counts number proper -colorings for each positive integer . The DP color D introduced by Kaul and Mudrock in 2019, a generalization with ? Let ? (resp. < denote property that = holds sufficiently large integers It an interesting problem finding graphs holds. showed if has even girth, then Thomason recently proved dominating vertex. We shall generalize their results this article. edge e let ? ? bridge be length shortest cycle containing otherwise. first show some However, converse statement conclusion fails infinitely many counterexamples. prove every contains spanning tree T such ? E ? odd contained C ? ? { } Some open problems are proposed