{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:06Z","timestamp":1753893846115,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","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 $L$ satisfies Hall's condition if for each subgraph $H$ of $G$, the inequality $|V(H)| \\leq \\sum_{\\sigma \\in \\mathcal{C}} \\alpha(H(\\sigma, L))$ is 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 $m$-completable for some $m \\geq \\chi(G)$ if every partial proper $m$-coloring of $G$ whose corresponding list assignment is Hall can be extended to a proper coloring of $G$.  In 2011, Bobga et al. posed the following questions:  (1) Are there examples of graphs that are Hall $m$-completable, but not Hall $(m+1)$-completable for some $m \\geq 3$? (2)  If $G$ is neither complete nor an odd cycle, is $G$ Hall $\\Delta(G)$-completable?  This paper establishes that for every $m \\geq 3$, there exists a graph that is Hall $m$-completable but not Hall $(m+1)$-completable and also that every bipartite planar graph $G$ is Hall $\\Delta(G)$-completable. <\/jats:p>","DOI":"10.37236\/4387","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:34:13Z","timestamp":1578699253000},"source":"Crossref","is-referenced-by-count":0,"title":["Completing Partial Proper Colorings using Hall's Condition"],"prefix":"10.37236","volume":"22","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":[[2015,7,1]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i3p6\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i3p6\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:14:36Z","timestamp":1579256076000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i3p6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,7,1]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2015,7,1]]}},"URL":"https:\/\/doi.org\/10.37236\/4387","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,7,1]]},"article-number":"P3.6"}}