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Packing chromatic number, ▫$(1,1,2,2)$▫-colorings, and characterizing the Petersen graph
Brešar, Boštjan ...
Pakirno kromatično število ▫$\chi_{\rho}(G)$▫ grafa ▫$G$▫ je najmanjše celo število ▫$k$▫, tako da lahko množico vozlišč grafa ▫$G$▫ razdelimo v množice ▫$\Pi_1,\ldots,\Pi_k$▫, kjer je ▫$\Pi_i$▫ (▫$i ...\in [k]$▫) ▫$i$▫-pakiranje. Postavljena in raziskovana je naslednja domneva: če je ▫$G$▫ podkubičen graf, potem je ▫$\chi_{\rho}(S(G))\le 5$▫, kjer je ▫$S(G)$▫ subdivizija grafa ▫$G$▫. Domneva je dokazana za vse posplošene prizme ciklov. Da dokažemo ta rezultat, je tudi dokazano, da če je ▫$G$▫ posplošena prizma cikla, potem je▫$G$▫ ▫$(1,1,2,2)$▫-obarvljiv natanko tedaj, ko ▫$G$▫ ni Petersenov graf. Veljavnost domneve je nadalje dokazana za grafe, ki jih lahko dobimo iz posplošenih prizm tako, da enega izmed dveh ▫$n$▫-ciklov nadomestimo z unijo ciklov, med katerimi je največ en 5-cikel. Obravnavano je tudi pakirno kromatično število grafov dobljenih s subdivizijo vsake izmed njegovih povezan enakokrat.
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