DP color functions versus chromatic polynomials (II)

For any connected graph G $G$ , let P ( m ) $P(G,m)$ and D ${P}_{DP}(G,m)$ denote the chromatic polynomial Dvořák Postle (DP) color function of respectively. It is known that ≤ ${P}_{DP}(G,m)\le P(G,m)$ holds for every positive integer $m$ . Let ≈ $D{P}_{\approx }$ (resp., < $D{P}_{\lt be set graphs which there exists an M $M$ such = ${P}_{DP}(G,m)=P(G,m)$ ${P}_{DP}(G,m)\lt all integers ≥ $m\ge M$ Determining sets important open problem on DP function. edge E 0 ${E}_{0}$ ℓ ${\ell }_{G}({E}_{0})$ size a shortest cycle C $C$ in ∣ ∩ $| E(C)\cap {E}_{0}| $ odd if exists, ∞ }_{G}({E}_{0})=\infty otherwise. We as e }_{G}(e)$ { } ${E}_{0}=\{e\}$ In this paper, we prove has spanning tree T $T$ each ∈ ⧹ $e\in E(G)\setminus E(T)$ edges $E(G)\setminus can labeled 1 2 … q ${e}_{1},{e}_{2},\ldots ,{e}_{q}$ with i + }_{G}({e}_{i})\le {\ell }_{G}({e}_{i+1})$ − $1\le i\le q-1$ ${e}_{i}$ contained ${C}_{i}$ }_{G}({e}_{i})$ ⊆ ∪ j : $E({C}_{i})\subseteq E(T)\cup \{{e}_{j}:1\le j\le i\}$ then As direct application, plane near-triangulations complete multipartite at least three partite belong to also show * ${E}^{* }_{G}({E}^{* })$ even satisfies certain conditions, belongs particular, 4 })=4$ where between two disjoint vertex subsets Both results extend ones by Dong Yang.