{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:22:18Z","timestamp":1759335738361,"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>The distinguishing number of a group $A$ acting faithfully on a set $X$, denoted $D(A,X)$, is the least number of colors needed to color the elements of $X$ so that no nonidentity element of $A$ preserves the coloring.  Given a map $M$ (an embedding of a graph in a closed surface) with vertex set $V$ and without loops or multiples edges, let $D(M)=D({\\rm Aut}(M),V)$, where ${\\rm Aut(M)}$ is the automorphism group of $M$; if $M$ is orientable, define $D^+(M)$ similarly, using only orientation-preserving automorphisms.  It is immediate that $D(M)\\leq 4$ and $D^+(M)\\leq 3$.  We use Russell and Sundaram's Motion Lemma to show that there are only finitely many maps $M$ with $D(M)&gt;2$. We show that if a group $A$ of automorphisms of a graph $G$ fixes no edges, then $D(A,V)=2$, with five exceptions.  That result is used to find the four maps with $D^+(M)=3$.  We also consider the distinguishing chromatic number $\\chi_D(M)$, where adjacent vertices get different colors.  We show $\\chi_D(M)\\leq \\chi(M)+3$ with equality in only finitely many cases, where $\\chi(M)$ is the chromatic number of the graph underlying $M$. We also show that $\\chi_D(M)\\leq 6$ for planar maps, answering a question of Collins and Trenk.  Finally, we discuss the implications for general group actions and give numerous problems for further study.<\/jats:p>","DOI":"10.37236\/537","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:52:56Z","timestamp":1578714776000},"source":"Crossref","is-referenced-by-count":15,"title":["Distinguishing Maps"],"prefix":"10.37236","volume":"18","author":[{"given":"Thomas W.","family":"Tucker","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,2,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p50\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p50\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:15:55Z","timestamp":1579302955000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p50"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2,28]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/537","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,2,28]]},"article-number":"P50"}}