Distinguishing labellings of group action on vector spaces and graphsKlavžar, Sandi (Avtor) Wong, Tsai-Lien (Avtor) Zhu, Xuding (Avtor) mathematicsgraph theorydistinguishing numbergroupgeneral linear groupvector spacegraphgraph automorphismdistinguishing setSuppose ▫$Gamma$▫ is a group acting on a set ▫$X$▫. A ▫$k$▫-labeling of ▫$X$▫ is a mapping ▫$c: to {1,2,...,k}$▫. A labeling ▫$c$▫ of ▫$X$▫ is distinguishing (with respect to the action of ▫$Gamma$▫) if for any ▫$g in Gamma$▫, ▫$g ne {mathrm{id}}_X$▫, there exists an element ▫$x in X$▫ such that ▫$c(x) ne c(g(x))▫$. The distinguishing number, ▫$D_Gamma(X)$▫, of the action of ▫$Gamma$▫ on ▫$X$▫ is the minimum ▫$k$▫ for which there is a ▫$k$▫-labeling which is distinguishing. This paper studies the distinguishing number of the linear group ▫$GL_n(K)$▫ over a field ▫$K$▫ acting on the vector space ▫$K^n$▫ and the distinguishing number of the automorphism group Aut▫$(G)$▫ of a graph ▫$G$▫ acting on ▫$V(G)$▫. The latter is called the distinguishing number of the graph ▫$G$▫ and is denoted by ▫$D(G)$▫. We determine the value of ▫$D_{GL_n(K)}(K^n)$▫ for all fields ▫$K$▫ and integers ▫$n$▫. For the distinguishing number of graphs, we study the possible value of the distinguishing number of a graph in terms of its automorphism group, its maximum degree, and other structure properties. It is proved that if ▫$mathrm{Aut}(G) = S_n$▫ and each orbit of Aut▫$(G)$▫ has size less than ▫$n choose n$▫, then ▫$D(G) = lceil n^{1/k} rceil$▫ for some positive integer ▫$k$▫. A Brooks type theorem for the distinguishing number is obtained: for any graph ▫$G$▫, ▫$D(G) le Delta(G)$▫, unless ▫$G$▫ is a complete graph, regular complete bipartite graph, or ▫$C_5$▫. We introduce the notion of uniquely distinguishable graphs and study the distinguishing number of disconnected graphs. ▫$Gamma$▫ deluje na množico ▫$X$▫. ▫$k$▫-označitev ▫$X$▫ je preslikava ▫$c: to {1,2,...,k}$▫. Označitev ▫$c$▫ množice ▫$X$▫ je razlikovalna (glede na delovanje ▫$Gamma$▫), če za vsak ▫$g in Gamma$▫, ▫$g ne {mathrm{id}}_X$▫ obstaja element ▫$x in X$▫, tako da je ▫$c(x) ne c(g(x))$▫. Razlikovalno število, ▫$D_Gamma(X)$▫, delovanja ▫$Gamma$▫ na ▫$X$▫, je najmanjši ▫$k$▫, za katerega obstaja ▫$k$▫-označitev, ki je razlikovalna. V tem članku študiramo razlikovalno število linearne grupe ▫$GL_n(K)$▫ nad poljem ▫$K$▫, ki deluje na vektorski prostor ▫$K^n$▫ in razlikovalno število grupe avtomorfizmov Aut▫$(G)$▫ grafa ▫$G$▫, ki deluje na ▫$V(G)$▫. Slednje je poimenovano razlikovalno število grafa ▫$G$▫ in označeno z ▫$D(G)$▫. V članku so določene vrednosti ▫$D_{GL_n(K)}(K^n)$▫ za vsa polja ▫$K$▫ in vsa števila ▫$n$▫. Glede razlikovalnega števila grafov študiramo možne vrednosti razlikovalnega števila grafa glede na njegovo grupo avtomorfizmov, njegovo največjo stopnjo in druge strukturne lastnosti. Dokazano je, da če je ▫$mathrm{Aut}(G) = S_n$▫ in ima vsaka orbita v Aut▫$(G)$▫ velikost manj kot ▫$n choose n$▫, tedaj je ▫$D(G) = lceil n^{1/k} rceil$▫ za neko naravno število ▫$k$▫. Dokazan je izrek Brooks-ovega tipa za razlikovalno število: za vsak graf ▫$G$▫ velja ▫$D(G) le Delta(G)$▫, razen če je ▫$G$▫ polni graf, regularni polni dovodelni graf, ali pa ▫$C_5$▫. Vpeljemo tudi pojem enolično razlikovalnih grafov in proučujemo razlikovalno število nepovezanih grafov.Elsevier20062015-07-10 14:54:55Znanstveno delo51559sl