{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:59Z","timestamp":1753893839142,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.37236\/266","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:24:59Z","timestamp":1578698699000},"source":"Crossref","is-referenced-by-count":2,"title":["A Note on $K_{\\Delta+1}^-$-Free Precolouring with $\\Delta$ Colours"],"prefix":"10.37236","volume":"16","author":[{"given":"Tom","family":"Rackham","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,9,18]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1n28\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1n28\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T21:40:40Z","timestamp":1579297240000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1n28"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,9,18]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/266","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,9,18]]},"article-number":"N28"}}