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arXiv · 2004.11173

Coloring Problems on Bipartite Graphs of Small Diameter

Victor A. Campos·Guilherme C. M. Gomes·Allen Ibiapina·Raul Lopes·Ignasi Sau·Ana Silva
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Abstract

We investigate a number of coloring problems restricted to bipartite graphs with bounded diameter. First, we investigate the $k$-List Coloring, List $k$-Coloring, and $k$-Precoloring Extension problems on bipartite graphs with diameter at most $d$, proving NP-completeness in most cases, and leaving open only the List $3$-Coloring and $3$-Precoloring Extension problems when $d=3$. Some of these results are obtained through a proof that the Surjective $C_6$-Homomorphism problem is NP-complete on bipartite graphs with diameter at most four. Although the latter result has been already proved [Vikas, 2017], we present ours as an alternative simpler one. As a byproduct, we also get that $3$-Biclique Partition is NP-complete. An attempt to prove this result was presented in [Fleischner, Mujuni, Paulusma, and Szeider, 2009], but there was a flaw in their proof, which we identify and discuss here. Finally, we prove that the $3$-Fall Coloring problem is NP-complete on bipartite graphs with diameter at most four, and prove that NP-completeness for diameter three would also imply NP-completeness of $3$-Precoloring Extension on diameter three, thus closing the previously mentioned open cases. This would also answer a question posed in [Kratochv\'il, Tuza, and Voigt, 2002].

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Victor A. Campos, Guilherme C. M. Gomes, Allen Ibiapina, Raul Lopes, Ignasi Sau, Ana Silva. 2020-04-23. Coloring Problems on Bipartite Graphs of Small Diameter. https://arxiv.org/abs/2004.11173

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