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Coloring graphs without short cycles and long induced paths.

Golovach, P.A. and Paulusma, D. and Song, J. (2014) 'Coloring graphs without short cycles and long induced paths.', Discrete applied mathematics., 167 . pp. 107-120.

Abstract

For an integer k≥1, a graph G is k-colorable if there exists a mapping c:VG→{1,…,k} such that c(u)≠c(v) whenever u and v are two adjacent vertices. For a fixed integer k≥1, the k-Coloring problem is that of testing whether a given graph is k-colorable. The girth of a graph G is the length of a shortest cycle in G. For any fixed g≥4 we determine a lower bound ℓ(g), such that every graph with girth at least g and with no induced path on ℓ(g) vertices is 3-colorable. We also show that for all fixed integers k,ℓ≥1, thek-Coloring problem can be solved in polynomial time for graphs with no induced cycle on four vertices and no induced path on ℓ vertices. As a consequence, for all fixed integers k,ℓ≥1 and g≥5, the k-Coloring problem can be solved in polynomial time for graphs with girth at least g and with no induced path on ℓ vertices. This result is best possible, as we prove the existence of an integer ℓ∗, such that already 4-Coloring is NP-complete for graphs with girth 4 and with no induced path on ℓ∗ vertices.

Item Type:Article
Keywords:Graph coloring, Girth, Forbidden induced subgraph
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.1016/j.dam.2013.12.008
Publisher statement:NOTICE: this is the author’s version of a work that was accepted for publication in Discrete applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Discrete applied mathematics, 167, 2014, 10.1016/j.dam.2013.12.008
Date accepted:No date available
Date deposited:06 January 2015
Date of first online publication:April 2014
Date first made open access:No date available

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