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Packing colorings of subcubic outerplanar graphsBrešar, Boštjan ; Gastineau, Nicolas ; Togni, OlivierZa dani graf ▫$G$▫ in nepadajoče zaporedje▫$S=(s_1,\ldots, s_k)$▫, katerega členi so naravna števila, preslikavi ▫$c:V(G)\longrightarrow \{1,\ldots,k\}$▫ pravimo ▫$S$▫-pakirno barvanje grafa ▫$G$▫, ... če je za vsaki različni vozlišči ▫$x$▫ in ▫$y$▫, ki pripadata ▫$c^{-1}(i)$▫, razdalja med ▫$x$▫ in ▫$y$▫ večja kot ▫$s_i$▫. Najmanjše tako število ▫$k$▫, da obstaja ▫$(1,2,\ldots,k)$▫-pakirno barvanje grafa ▫$G$▫, imenujemo pakirno kromatično število grafa ▫$G$▫ in ga označimo z ▫$\chi_{\rho}(G)$▫. Vprašanje omejenosti pakirnega kromatičnega števila v razredu podkubičnih (ravninskih) grafov je bilo obravnavano v več prejšnih člankih; pred kratkim je bilo ugotovljeno, da je invarianta neomejena v razredu vseh podkubičnih grafov. V članku dokažemo, da je pakirno kromatično število vsakega 2-povezanega, dvodelnega, podkubičnega, zunanje ravninskega grafa omejeno s ▫$7$▫. Dokažemo tudi, da vsak podkubičen, zunanje ravninski graf brez trikotnika premore ▫$(1,2,2,2)$▫-pakirno barvanje in da obstaja podkubičen, zunanje ravninski graf s trikotnikom, ki takšnega pakirnega barvanja ne premore. Poleg tega ugotovimo, da obstaja podkubičen, zunanje ravninski graf brez trikotnika, ki ne premore ▫$(1,2,2,3)$▫-pakirnega barvanja. Podobno dihotomijo predstavimo za dvodelne zunanje ravninske grafe; vsak tak graf premore ▫$S$▫-pakirno barvanje za ▫$S=(1,3,\ldots,3)$▫, kjer se ▫$3$▫ pojavi ▫$\Delta$▫-krat (pri čemer je ▫$\Delta$▫ največja stopnja vozlišč grafa) in ta lastnost ne velja, če eno od števil ▫$3$▫ v zaporedju zamenjamo s ▫$4$▫.Source: Aequationes mathematicae. - ISSN 0001-9054 (Vol. 94, iss. 5, Oct. 2020, 945-967)Type of material - article, component part ; adult, seriousPublish date - 2020Language - englishCOBISS.SI-ID - 27467011
Author
Brešar, Boštjan |
Gastineau, Nicolas |
Togni, Olivier
Topics
zunanje ravninski graf |
pakirno kromatično število |
kubičen graf |
barvanje |
pakiranje |
outerplanar graph |
packing chromatic number |
cubic graph |
coloring |
packing
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Brešar, Boštjan | 17005 |
Gastineau, Nicolas | |
Togni, Olivier |
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