The dichromatic number of digraphs without induced subdigraphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2025-08-14 DOI:10.1016/j.disc.2025.114729

Bin Chen , Xinmin Hou

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引用次数: 0

Abstract

Let D be a digraph. The dichromatic number of D is the smallest number of colors needed to color the vertices of D such that each color class induces a subdigraph without directed cycles. In this paper, we investigate a conjecture proposed by Aboulker, Charbit and Naserasr, which extends the well known Gyárfás-Sumner conjecture to digraphs. Let

\vec{P_{k}}

and

\vec{C_{k}}

be a directed path and a directed cycle on k vertices, respectively. Denote by

C_{3}

the family of all oriented cycles on 3 vertices. We prove that every

{\vec{P_{7}}, \vec{C_{4}}, C_{3}}

-free oriented graph has dichromatic number at most 190. Additionally, we verify that the dichromatic number of any

{\vec{P_{6}}, C_{3}}

-free oriented graph is at most 178, improving a result of Aboulker, Aubian, Charbit and Thomassé.

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没有诱导子向图的有向图的二色数

设D是有向图。D的二色数是为D的顶点上色所需的最小颜色数，这样每个颜色类都能产生一个没有有向环的子图。本文研究了Aboulker， Charbit和Naserasr提出的一个猜想，它将众所周知的Gyárfás-Sumner猜想扩展到有向图。设Pk→和Ck→分别是k个顶点上的有向路径和有向循环。用C3表示3个顶点上所有有向环的族。证明了每一个无{P7→，C4→，C3}取向图的二色数不超过190。此外，我们验证了任意无{P6→，C3}取向图的二色数最多为178，改进了Aboulker， Aubian， Charbit和thomass的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.