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Journal Articles Journal of Graph Theory Year : 2013

A Complexity Dichotomy for the Coloring of Sparse Graphs

Louis Esperet
Mickaël Montassier
Pascal Ochem

Abstract

Gallucio, Goddyn and Hell proved in 2001 that in any minor-closed class, graphs with large enough girth have a homomorphism to any given odd cycle. In this paper, we study the computational aspects of this problem. We show that for any minor-closed class $F$ containing all planar graphs, and such that all minimal forbidden minors are 3-connected, the following holds: for any $k$ there is a $g = g(F, k)$ such that every graph of girth at least $g$ in $F$ has a homomorphism to $C_2k+1$, but deciding whether a graph of girth $g-1$ in $F$ has a homomorphism to $C_2k+1$ is NP-complete. The classes of graphs on which this result applies include planar graphs, $K_n$-minor free graphs, and graphs with bounded Colin de Verdière parameter (for instance, linklessly embeddable graphs). We also show that the same dichotomy occurs in problems related to a question of Havel (1969) and a conjecture of Steinberg (1976) about the 3-colorability of sparse planar graphs.

Dates and versions

lirmm-00736468 , version 1 (28-09-2012)

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Cite

Louis Esperet, Mickaël Montassier, Pascal Ochem, Alexandre Pinlou. A Complexity Dichotomy for the Coloring of Sparse Graphs. Journal of Graph Theory, 2013, 73 (1), pp.85-102. ⟨10.1002/jgt.21659⟩. ⟨lirmm-00736468⟩
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