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Coloring Problems on Bipartite Graphs of Small Diameter
- Publication Year :
- 2020
-
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].<br />Comment: 21 pages, 9 figures
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2004.11173
- Document Type :
- Working Paper