Packing coloring of hypercubes with extended Hamming codes

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics Pub Date : 2024-08-20 DOI:10.1016/j.dam.2024.07.048

Petr Gregor , Jaka Kranjc , Borut Lužar , Kenny Štorgel

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引用次数: 0

Abstract

A packing $k$ -coloring of a graph $G$ is a mapping assigning a positive integer (a color) from the set $(1, \dots, k)$ to every vertex of $G$ such that every two distinct vertices of color $c$ are at distance at least $c + 1$ . The minimum value $k$ such that $G$ admits a packing $k$ -coloring is called the packing chromatic number of $G$ . In this paper, we continue the study of the packing chromatic number of hypercubes and we improve the upper bounds reported by Torres and Valencia-Pabon (2015) by presenting recursive constructions of subsets of distant vertices making use of the properties of the extended Hamming codes. We also answer in negative a question on the packing chromatic number of Cartesian products raised by Brešar et al. (2007).

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用扩展汉明码对超立方体进行打包着色

图 G 的打包 k 着色是给 G 的每个顶点从集合 1，...，k 中指定一个正整数（一种颜色）的映射，使得颜色 c 的每两个不同顶点的距离至少为 c+1。本文将继续研究超立方体的堆积色度数，并利用扩展汉明码的特性，提出了远距离顶点子集的递归构造，从而改进了 Torres 和 Valencia-Pabon (2015) 报告的上界。我们还否定地回答了 Brešar 等人（2007 年）提出的关于笛卡尔积的包装色度数的问题。

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 Applied Mathematics 数学-应用数学

CiteScore

2.30

自引率

9.10%

发文量

422

审稿时长

4.5 months

期刊介绍： The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal. Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.