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

Let $G$ be a simple graph of maximum degree $\Delta \geq 3$, not containing $K_{\Delta + 1}$, and ${\cal L}$ a list assignment to $V(G)$ such that $|{\cal L}(v)| = \Delta$ for all $v \in V(G)$.  Given a set $P \subset V(G)$ of pairwise distance at least $d$ then Albertson, Kostochka and West (2004) and Axenovich (2003) have shown that every ${\cal L}$-precolouring of $P$ extends to a ${\cal L}$-colouring of $G$ provided $d \geq 8$. Let $K_{\Delta + 1}^-$ denote the graph $K_{\Delta + 1}$ with one edge removed.  In this paper, we consider the problem above and the effect on the pairwise distance required when we additionally forbid either $K_{\Delta + 1}^-$ or $K_{\Delta}$ as a subgraph of $G$.  We have the corollary that an extra assumption of 3-edge-connectivity in the above result is sufficient to reduce this distance from $8$ to $4$. This bound is sharp with respect to both the connectivity and distance.  In particular, this corrects the results of Voigt (2007, 2008) for which counterexamples are given.

A Note on $K_{\Delta+1}^-$-Free Precolouring with $\Delta$ Colours