| Title: | Distinguishing labellings of group action on vector spaces and graphs |
|---|
| Authors: | ID Klavžar, Sandi (Author)
ID Wong, Tsai-Lien (Author)
ID Zhu, Xuding (Author) |
|---|
| Files: |  http://dx.doi.org/10.1016/j.jalgebra.2006.01.045
|
|---|
| Language: | English |
|---|
| Work type: | Scientific work |
|---|
| Typology: | 1.01 - Original Scientific Article |
|---|
| Organization: | PEF - Faculty of Education
|
|---|
| Abstract: | Suppose ▫$Gamma$▫ 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. |
|---|
| Keywords: | mathematics, graph theory, distinguishing number, group, general linear group, vector space, graph, graph automorphism, distinguishing set |
|---|
| Publication status: | Published |
|---|
| Publication version: | Version of Record |
|---|
| Publisher: | Elsevier |
|---|
| Year of publishing: | 2006 |
|---|
| Number of pages: | str. 626-641 |
|---|
| Numbering: | Letn. 303, št. 2 |
|---|
| PID: | 20.500.12556/DKUM-51559  |
|---|
| UDC: | 519.17:512.54 |
|---|
| ISSN on article: | 0021-8693 |
|---|
| COBISS.SI-ID: | 14075225 |
|---|
| NUK URN: | URN:SI:UM:DK:BPQHR2JK |
|---|
| Publication date in DKUM: | 10.07.2015 |
|---|
| Views: | 1478 |
|---|
| Downloads: | 104 |
|---|
| Metadata: |    |
|---|
| Categories: | Misc.
|
|---|
|
:
|
Copy citation |
|---|
| | | | Average score: | (0 votes) |
|---|
| Your score: | Voting is allowed only for logged in users. |
|---|
| Share: |  |
|---|
Hover the mouse pointer over a document title to show the abstract or click
on the title to get all document metadata. |
