{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:45Z","timestamp":1753893825702,"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>Let $G$ be a connected graph with maximum degree $\\Delta$. Brooks' theorem states that $G$ has a $\\Delta$-coloring unless $G$ is a complete graph or an odd cycle. A graph $G$ is degree-choosable\u00a0if $G$ can be properly colored from its lists whenever each vertex $v$ gets a list of $d(v)$ colors. In the context of list coloring, Brooks' theorem can be strengthened to the following. Every connected graph $G$ is degree-choosable unless each block of $G$ is a complete graph or an odd cycle; such a graph $G$ is a Gallai tree.  This degree-choosability result was further strengthened to Alon\u2014Tarsi orientations;\u00a0these are orientations of $G$ in which the number of spanning Eulerian\u00a0subgraphs with an even number of edges differs from the number with an odd\u00a0number of edges. A graph $G$ is degree-AT\u00a0if $G$ has an Alon\u2014Tarsi\u00a0orientation in which each vertex has indegree at least 1. Alon and Tarsi showed that if $G$ is degree-AT, then $G$ is also\u00a0degree-choosable. Hladk\u00fd, Kr\u00e1l', and\u00a0Schauz showed that a connected graph is degree-AT if and only if it is not a\u00a0Gallai tree. In this paper, we consider pairs $(G,x)$ where $G$ is a connected\u00a0graph and $x$ is some specified vertex in $V(G)$. We characterize pairs such\u00a0that $G$ has no Alon\u2014Tarsi orientation in which each vertex has indegree at\u00a0least 1 and $x$ has indegree at least 2. When $G$ is 2-connected, the\u00a0characterization is simple to state.<\/jats:p>","DOI":"10.37236\/6179","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:50:24Z","timestamp":1578671424000},"source":"Crossref","is-referenced-by-count":0,"title":["Beyond Degree Choosability"],"prefix":"10.37236","volume":"24","author":[{"given":"Daniel W.","family":"Cranston","sequence":"first","affiliation":[]},{"given":"Landon","family":"Rabern","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,8,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p29\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p29\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:50:23Z","timestamp":1579236623000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i3p29"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,11]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/6179","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,8,11]]},"article-number":"P3.29"}}