Transitive and Gallai colorings of the complete graph

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics Pub Date : 2025-08-21 DOI:10.1016/j.ejc.2025.104225

Ron M. Adin , Arkady Berenstein , Jacob Greenstein , Jian-Rong Li , Avichai Marmor , Yuval Roichman

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引用次数: 0

Abstract

A Gallai coloring of the complete graph is an edge-coloring with no rainbow triangle. This concept first appeared in the study of incomparability graphs and anti-Ramsey theory. A directed analogue, called transitive coloring, was introduced by Berenstein, Greenstein and Li in a rather general setting. It is studied here for the acyclic tournament. The interplay of the two notions yields new enumerative results and algebraic perspectives.

We first count Gallai and transitive colorings of the complete graph which use the maximal number of colors. The quasisymmetric generating functions of these colorings, equipped with a natural descent set, are shown to be Schur-positive for any number of colors. Explicit Schur expansions are described when the number of colors is maximal. It follows that descent sets of maximal Gallai and transitive colorings are equidistributed with descent sets of perfect matchings and pattern-avoiding indecomposable permutations, respectively.

Corresponding commutative algebras are also studied. Their dimensions are shown to be equal to the number of Gallai colorings of the complete graph and the number of transitive colorings of the acyclic tournament, respectively. Relations to Orlik-Terao algebras are established.

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完全图的传递着色和盖莱着色

完全图的盖莱着色是一种没有彩虹三角形的边着色。这个概念最早出现在不可比较图和反拉姆齐理论的研究中。Berenstein， Greenstein和Li在一般情况下引入了一种称为传递着色的定向类似物。这是为无环锦标赛研究的。这两个概念的相互作用产生了新的枚举结果和代数观点。我们首先计算了完全图中使用最大颜色数的盖勒着色和传递着色。具有自然下降集的这些着色的拟对称生成函数对于任意数量的颜色都是schur正的。当颜色数量达到最大值时，描述显式舒尔展开。由此可知，极大加勒着色和传递着色的下降集分别与完美匹配和避免模式不可分解置换的下降集是等分布的。并研究了相应的交换代数。它们的维数分别等于完全图的加勒着色的个数和无环比赛场的传递着色的个数。建立了与orlikterao代数的关系。

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

European Journal of Combinatorics 数学-数学

CiteScore

2.10

自引率

10.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.