{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:18Z","timestamp":1753893798086,"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>Collins and Trenk define the distinguishing chromatic number $\\chi_D(G)$ of a graph $G$ to be the minimum number of colors needed to properly color the vertices of $G$ so that the only automorphism of $G$ that preserves colors is the identity.  They prove results about $\\chi_D(G)$ based on the underlying graph $G$.  In this paper we prove results that relate $\\chi_D(G)$ to the automorphism group of $G$.  We prove two upper bounds for $\\chi_D(G)$ in terms of the chromatic number $\\chi(G)$ and show that each result is tight: (1) if Aut$(G)$ is any finite group of order $p_1^{i_1} p_2^{i_2} \\cdots p_k^{i_k}$ then $\\chi_D(G) \\le \\chi(G) + i_1 + i_2 \\cdots + i_k$, and (2) if Aut$(G)$ is a finite and abelian group written Aut$(G) = {\\Bbb Z}_{p_{1}^{i_{1}}}\\times \\cdots \\times {\\Bbb Z}_{p_{k}^{i_{k}}}$ then we get the improved bound $\\chi_D(G) \\le \\chi(G) + k$.  In addition, we characterize automorphism groups of graphs with $\\chi_D(G) = 2$ and discuss similar results for graphs with $\\chi_D(G)=3$.<\/jats:p>","DOI":"10.37236\/177","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:28:17Z","timestamp":1578716897000},"source":"Crossref","is-referenced-by-count":7,"title":["Bounds on the Distinguishing Chromatic Number"],"prefix":"10.37236","volume":"16","author":[{"given":"Karen L.","family":"Collins","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mark","family":"Hovey","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ann N.","family":"Trenk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2009,7,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r88\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r88\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T02:50:22Z","timestamp":1579315822000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r88"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,7,24]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/177","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,7,24]]},"article-number":"R88"}}