AI 中文总结
本研究发现Aboulker等人隐含的简单归约可直接由原始图论Hajnal–Szemerédi定理证明Treglown猜想及Czygrinow等人的有向图相关结果，修改归约后还得到了此类有向图公平无环$k$着色的多项式时间算法。
AI 中文摘要
Treglown以互补形式猜想：对任意正整数$k$，若有向图$D$满足对所有顶点$v\in V(D)$都有$\min\{d^+(v),d^-(v)\}\le k-1$，则$D$存在公平无环$k$着色。若该猜想成立，将可推导出Czygrinow、DeBiasio、Kierstead和Molla证明的有向图Hajnal-Szemerédi定理的无环着色版本（该结果又可推导出原始的图论Hajnal-Szemerédi定理）。\n 研究发现，Aboulker、Oijid、Petit、Rocton和Simon的工作中隐含着一种简单归约，该归约出人意料地表明Treglown的猜想（以及Czygrinow等人的结果）可直接由原始的图论Hajnal–Szemerédi定理推导得出。我们对该归约稍作修改，证明存在多项式时间算法，可在此类有向图中找到公平无环$k$着色。
英文摘要
Treglown conjectured (in a complementary form) that for every positive integer $k$, every digraph $D$ satisfying $\min\{d^+(v),d^-(v)\}\le k-1$ for all $v\in V(D)$ has an equitable acyclic $k$-coloring. If true, this would imply the acyclic coloring versions of the Hajnal-Szemerédi theorem for digraphs proved by Czygrinow, DeBiasio, Kierstead, and Molla (which in turn imply the original Hajnal-Szemerédi theorem for graphs). As it turns out, there is a simple reduction implicit in Aboulker, Oijid, Petit, Rocton, and Simon which surprisingly shows that Treglown's conjecture (and thus the results of Czygrinow, DeBiasio, Kierstead, and Molla) follows directly from the original Hajnal--Szemerédi theorem for graphs. We slightly modify the reduction in order to show that there exists a polynomial time algorithm for finding an equitable acyclic $k$-coloring in such a digraph.
Comments6 pages
