English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The DP-coloring is a generalization of the list coloring, introduced byDvo\v{r}\'{a}k and Postle. Let $\mathcal{H}=(L,H)$ be a cover of a graph $G$,$P_{DP}(G,\mathcal{H})$ be the number of $\mathcal{H}$-colorings of $G$. The DPcolor function $P_{DP}(G,m)$ of $G$, introduced by Kaul and Mudrock, is theminimum value of $P_{DP}(G,\mathcal{H})$ where the minimum is taken over allpossible $m$-fold covers $\mathcal{H}$ of $G$. For the family of $n$-vertexconnected graphs, one can deduce that trees maximize the DP color function,from two results of Kaul and Mudrock. In this paper we obtain tight upperbounds for the DP color function of $n$-vertex $2$-connected graphs. Anotherconcern in this paper is the canonical labeling in a cover. It is well knownthat if a $m$-fold cover $\mathcal{H}$ of a graph $G$ has a canonical labeling,then $P_{DP}(G,\mathcal{H})=P(G,m)$ in which $P(G,m)$ is the chromaticpolynomial of $G$. However the converse statement of this conclusion is notalways true. We give examples that for some $m$ and $G$,$P_{DP}(G,\mathcal{H})=P(G,m)$, but $\mathcal{H}$ has no canonical labelings.We also prove that when $G$ is a unicycle graph or a theta graph, for each$m\geq 3$, if $P_{DP}(G,\mathcal{H})=P(G,m)$, then $\mathcal{H}$ has acanonical labeling.

Bounds for DP color function and canonical labelings