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Distinguishing graphs by total colouringsKalinowski, Rafał ; Pilśniak, Monika ; Woźniak, Mariusz, 1951-Vpeljemo totalno razlikovalno število ▫$D^{\prime\prime}(G)$▫ grafa ▫$G$▫ kot najmanjše takšno število ▫$d$▫, da ima ▫$G$▫ totalno barvanje (ne nujno pravilno) z ▫$d$▫ barvami, ki ga ohranja samo ... trivialni avtomorfizem. Gre za analogijo pojmov razlikovalnega števila ▫$D(G)$▫ in razlikovalnega indeksa ▫$D^\prime(G)$▫, ki sta definirana za barvanja vozlišč oziroma povezav. Dobimo splošno ostro zgornjo mejo: ▫$D^{\prime\prime}(G)\leq \lceil\sqrt{\Delta(G)}\rceil$▫ za vsak povezan graf ▫$G$▫. Vpeljemo tudi totalno razlikovalno kromatsko število ▫$\chi^{\prime\prime}_D(G)$▫ definirano podobno za pravilna totalna barvanja grafa ▫$G$▫. Dokažemo, da je ▫$\chi^{\prime\prime}_{D}(G) \leq \chi^{\prime\prime}(G) + 1$▫ za vsak povezan graf ▫$G$▫ s totalnim kromatskim številom ▫$\chi^{\prime\prime}(G)$▫. Še več, če je ▫$\chi^{\prime\prime}(G) \geq\Delta(G) + 2$▫, potem je ▫$\chi^{\prime\prime}_{D}(G) = \chi(G)$▫. Te rezultate dokažemo tako da določimo ostre zgornje meje za minimalno število barv v pravilnem totalnem barvanju, v katerem iz vsakega vozlišča izhaja drugačna množica barvnih poti.Source: Ars mathematica contemporanea. - ISSN 1855-3966 (Vol. 11, no. 1, 2016, str. 79-89)Type of material - article, component part ; adult, seriousPublish date - 2016Language - englishCOBISS.SI-ID - 17841241
Author
Kalinowski, Rafał |
Pilśniak, Monika |
Woźniak, Mariusz, 1951-
Topics
totalna barvanja grafov |
zlom simetrije v grafih |
totalno razlikovalno število |
totalno razlikovalno kromatsko število |
total colourings of graphs |
symmetry breaking in graphs |
total distinguishing number |
total distinguishing chromatic number
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Kalinowski, Rafał | ![]() |
Pilśniak, Monika | ![]() |
Woźniak, Mariusz, 1951- | ![]() |
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