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].
翻译:首先,我们调查在直径以美元为单位的双面图上出现的一些彩色问题。首先,我们调查了在直径以美元最多为单位的双面图上出现的以美元计价的彩色、以美元计价的彩色和以美元计价的彩色扩展问题。首先,我们调查了以美元计价的彩色扩展问题。首先,我们调查了以美元计价的彩色扩展问题。首先,我们调查了在直径以美元计价的双面图上出现的以美元计价、以美元计价的彩色扩展问题和以美元计价的彩色扩展问题。尽管后者的结果已经证明[2017年,维卡],但我们作为替代的更简单的一个问题展示了我们的彩色扩展问题。作为副产品,我们还只看到3美元-纸色分割和3美元的彩色扩展扩展问题。试图证明,[Fleischner、Mujuni、Paulusma和Szeidar,2009年],但其中有一个证据有缺陷,我们在这里确定和讨论的彩色最直径直径的三度问题。最后证明,我们证明了的彩度将证明。