Coloring Problems on Bipartite Graphs of Small Diameter
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 NPcompleteness 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 NPcomplete 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 NPcomplete. 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 NPcomplete on bipartite graphs with diameter at most four, and prove that NPcompleteness for diameter three would also imply NPcompleteness of $3$Precoloring Extension on diameter three, thus closing the previously mentioned open cases. This would also answer a question posed in [Kratochvíl, Tuza, and Voigt, 2002].
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.11173
 Bibcode:
 2020arXiv200411173C
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms;
 05C15;
 G.2.2;
 F.2.2
 EPrint:
 21 pages, 9 figures