{"title":"Achromatic colorings of polarity graphs","authors":"Vladislav Taranchuk , Craig Timmons","doi":"10.1016/j.ffa.2024.102497","DOIUrl":null,"url":null,"abstract":"<div><p>A complete partition of a graph <em>G</em> is a partition of the vertex set such that there is at least one edge between any two parts. The largest <em>r</em> such that <em>G</em> has a complete partition into <em>r</em> parts, each of which is an independent set, is the achromatic number of <em>G</em>. We determine the achromatic number of polarity graphs of biaffine planes coming from generalized polygons. Our colorings of a family of unitary polarity graphs are used to solve a problem of Axenovich and Martin on complete partitions of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free graphs. Furthermore, these colorings prove that there are sequences of graphs which are optimally complete and have unbounded degree, a problem that had been studied for the sequence of hypercubes independently by Roichman, and Ahlswede, Bezrukov, Blokhuis, Metsch, and Moorhouse.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102497"},"PeriodicalIF":1.2000,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1071579724001369","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A complete partition of a graph G is a partition of the vertex set such that there is at least one edge between any two parts. The largest r such that G has a complete partition into r parts, each of which is an independent set, is the achromatic number of G. We determine the achromatic number of polarity graphs of biaffine planes coming from generalized polygons. Our colorings of a family of unitary polarity graphs are used to solve a problem of Axenovich and Martin on complete partitions of -free graphs. Furthermore, these colorings prove that there are sequences of graphs which are optimally complete and have unbounded degree, a problem that had been studied for the sequence of hypercubes independently by Roichman, and Ahlswede, Bezrukov, Blokhuis, Metsch, and Moorhouse.
图 G 的完整分割是顶点集的分割，使得任意两部分之间至少有一条边。我们确定了来自广义多边形的双折线平面极性图的消色数。我们对单元极性图族的着色用于解决阿克森诺维奇和马丁关于无 C4 图的完全分割的问题。此外，这些着色证明了存在最优完整且度无界的图序列，这个问题曾由罗伊克曼、阿尔斯韦德、贝兹鲁科夫、布洛克胡斯、梅奇和穆尔豪斯独立研究过超立方体序列。
期刊介绍：
Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering.
For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods.
The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.
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