{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:34Z","timestamp":1753893814529,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In the context of list coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list coloring. The graph $G$ with list assignment $L$, abbreviated $(G,L)$, satisfies Hall's condition if for each subgraph $H$ of $G$, the inequality $|V(H)| \\leq \\sum_{\\sigma \\in \\mathcal{C}} \\alpha(H(\\sigma, L))$\u00a0is satisfied, where $\\mathcal{C}$ is the set of colors and $\\alpha(H(\\sigma, L))$ is the independence number of the subgraph of $H$ induced on the set of vertices having color $\\sigma$ in their lists. A list assignment $L$ to a graph $G$ is called Hall if $(G,L)$ satisfies Hall's condition. A graph $G$ is Hall $k$-extendible for some $k \\geq \\chi(G)$ if every $k$-precoloring of $G$ whose corresponding list assignment is Hall can be extended to a proper $k$-coloring of $G$. In 2011, Bobga et al. posed the question: If $G$ is neither complete nor an odd cycle, is $G$ Hall $\\Delta(G)$-extendible? This paper establishes an affirmative answer to this question: every graph $G$ is Hall $\\Delta(G)$-extendible. Results relating to the behavior of Hall extendibility under subgraph containment are also given. Finally, for certain graph families, the complete spectrum of values of $k$ for which they are Hall $k$-extendible is presented. We include a focus on graphs which are Hall $k$-extendible for all $k \\geq \\chi(G)$, since these are graphs for which satisfying the obviously necessary Hall's condition is also sufficient for a precoloring to be extendible.<\/jats:p>","DOI":"10.37236\/5687","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T19:56:47Z","timestamp":1578686207000},"source":"Crossref","is-referenced-by-count":0,"title":["Every Graph $G$ is Hall $\\Delta(G)$-Extendible"],"prefix":"10.37236","volume":"23","author":[{"given":"Sarah","family":"Holliday","sequence":"first","affiliation":[]},{"given":"Jennifer","family":"Vandenbussche","sequence":"additional","affiliation":[]},{"given":"Erik E.","family":"Westlund","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,11,10]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i4p19\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i4p19\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:10:47Z","timestamp":1579237847000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i4p19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,11,10]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,10,14]]}},"URL":"https:\/\/doi.org\/10.37236\/5687","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,11,10]]},"article-number":"P4.19"}}