Colorings and Homomorphisms of Minor Closed Classes
We relate acyclic (and star) chromatic number of a graph to the chromatic number of its minors and as a consequence we show that the set of all triangle free planar graphs is homomorphism bounded by a triangle free graph. This solves a problem posed in []. It also improves the best known bound for the star chromatic number of planar graphs from 80 to 30. Our method generalizes to all minor closed classes and puts Hadwiger conjecture in yet another context.
KeywordsPlanar Graph Chromatic Number Proper Coloring Mixed Graph Acyclic Orientation
Unable to display preview. Download preview PDF.
- P. Dreyer, Ch. Malon, J. Ne¡¦set¡¦ril: Universal H-colorable graphs without a given configuration, Discrete Math.(in press)Google Scholar
- G. Fertin, A. Raspaud, B. Reed: On Star Coloring of graphs. In: Proccedings of GW’01, LNCS, Springer VerlagGoogle Scholar
- J. Fiala, J. Kratochv´ıl, A. Proskurowski: Partial covers of graphs, to appear in Discussiones Mathematicae Graph Theory.Google Scholar
- 8.S.L. Hakimi: On the degree of the vertices of a directed graph, J. Franklin Inst. 279 (1965), 4.Google Scholar
- 10.T. Jensen, B. Toft: Graph coloring problems, Willey, 1995.Google Scholar
- 11.A. Kostochka: On the minimum of the Hadwiger number for graphs with given average degree, Metody Diskret. Analiz., 38(1982), Novosibirsk, 37–58, in Russian,English translation: AMS Translations (2), 132(1986), 37–58.Google Scholar
- T. H. Marshall, R. Nasraser, J. Ne¡¦set¡¦ril: Homomorphism Bounded Classes of Graphs (to appear in European J. Comb.)Google Scholar
- J. Nešetřil, P. Ossona de Mendez: Foldings (submitted)Google Scholar