Coloring immersion-free graphs

A graph H is immersed in a graph G if the vertices of H are mapped to (distinct) vertices of G, and the edges of H are mapped to paths joining the corresponding pairs of vertices of G, in such a way that the paths are pairwise edge-disjoint. The notion of an immersion is quite similar to the well-kn...

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Veröffentlicht in:Journal of combinatorial theory. Series B Jg. 121; S. 284 - 307
Hauptverfasser: Kakimura, Naonori, Kawarabayashi, Ken-ichi
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Inc 01.11.2016
Schlagworte:
Coloring
Fixed-parameter algorithm
Immersion
Coloring
Fixed-parameter algorithm
Immersion
ISSN:0095-8956, 1096-0902
Online-Zugang:Volltext
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Zusammenfassung:A graph H is immersed in a graph G if the vertices of H are mapped to (distinct) vertices of G, and the edges of H are mapped to paths joining the corresponding pairs of vertices of G, in such a way that the paths are pairwise edge-disjoint. The notion of an immersion is quite similar to the well-known notion of a minor, as structural approach inspired by the theory of graph minors has been extremely successful in immersions. Hadwiger's conjecture on graph coloring, generalizing the Four Color Theorem, states that every loopless graph without a Kk-minor is (k−1)-colorable, where Kk is the complete graph on k vertices. This is a long standing open problem in graph theory, and it is even unknown whether it is possible to determine ck-colorability of Kk-minor-free graphs in polynomial time for some constant c. In this paper, we address coloring graphs without H-immersion. In contrast to coloring H-minor-free graphs, we show the following:1.there exists a fixed-parameter algorithm to decide whether or not a given graph G without an immersion of a graph H of maximum degree d is (d−1)-colorable, where the size of H is a parameter. In fact, if G is (d−1)-colorable, the algorithm produces such a coloring, and2.for any positive integer k (k≥6), it is NP-complete to decide whether or not a given graph G without a Kk-immersion is (k−3)-colorable.
ISSN:0095-8956
1096-0902
DOI:10.1016/j.jctb.2016.07.005