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DP color functions versus chromatic polynomials (II)

For any connected graph $G$, let $P(G,m)$ and $P_{DP}(G,m)$ denote the chromatic polynomial and DP color function of $G$, respectively. It is known that $P_{DP}(G,m)\le P(G,m)$ holds for every positive integer $m$. Let $DP_\approx$ (resp. $DP_<$) be the set of graphs $G$ for which there exists an integer $M$ such that $P_{DP}(G,m)=P(G,m)$ (resp. $P_{DP}(G,m)<P(G,m)$) holds for all integers $m \ge M$. Determining the sets $DP_\approx$ and $DP_<$ is a key problem on the study of the DP color function. For any edge set $E_0$ of $G$, let $\ell_G(E_0)$ be the length of a shortest cycle $C$ in $G$ such that $|E(C)\cap E_0|$ is odd whenever such a cycle exists, and $\ell_G(E_0)=\infty$ otherwise. Write $\ell_G(E_0)$ as $\ell_G(e)$ if $E_0=\{e\}$. In this paper, we prove that if $G$ has a spanning tree $T$ such that $\ell_G(e)$ is odd for each $e\in E(G)\setminus E(T)$, the edges in $E(G)\setminus E(T)$ can be labeled as $e_1,e_2,\cdots, e_q$ with $\ell_G(e_i)\le \ell_G(e_{i+1})$ for all $1\le i\le q-1$ and each edge $e_i$ is contained in a cycle $C_i$ of length $\ell_G(e_i)$ with $E(C_i)\subseteq E(T)\cup \{e_j: 1\le j\le i\}$, then $G$ is a graph in $DP_{\approx}$. As a direct application of this conclusion, all plane near-triangulations and complete multipartite graphs with at least three partite sets belong to $DP_{\approx}$. We also show that if $E^*$ is an edge set of $G$ such that $\ell_{G}(E^*)$ is even and $E^*$ satisfies certain conditions, then $G$ belongs to $DP_<$. In particular, if $\ell_G(E^*)=4$, where $E^*$ is a set of edges between two disjoint vertex subsets of $G$, then $G$ belongs to $DP_<$. Both results extend known ones in [DP color functions versus chromatic polynomials, $Advances\ in\ Applied\ Mathematics$ 134 (2022), article 102301].