{"title":"Spectral upper bounds for the Grundy number of a graph","authors":"Thiago Assis, Gabriel Coutinho, Emanuel Juliano","doi":"10.1016/j.disc.2024.114326","DOIUrl":null,"url":null,"abstract":"<div><div>The Grundy number of a graph is the minimum number of colors needed to properly color the graph using the first-fit greedy algorithm regardless of the initial vertex ordering. Computing the Grundy number of a graph is an NP-Hard problem. There is a characterization in terms of induced subgraphs: a graph has a Grundy number at least k if and only if it contains a <em>k</em>-atom. In this paper, using properties of the matching polynomial, we determine the smallest possible largest eigenvalue of a <em>k</em>-atom. With this result, we present an upper bound for the Grundy number of a graph in terms of the largest eigenvalue of its adjacency matrix. We also present another upper bound using the largest eigenvalue and the size of the graph. Our bounds are asymptotically tight for some infinite families of graphs and provide improvements on the known bounds for the Grundy number of sparse random graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 3","pages":"Article 114326"},"PeriodicalIF":0.7000,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004576","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The Grundy number of a graph is the minimum number of colors needed to properly color the graph using the first-fit greedy algorithm regardless of the initial vertex ordering. Computing the Grundy number of a graph is an NP-Hard problem. There is a characterization in terms of induced subgraphs: a graph has a Grundy number at least k if and only if it contains a k-atom. In this paper, using properties of the matching polynomial, we determine the smallest possible largest eigenvalue of a k-atom. With this result, we present an upper bound for the Grundy number of a graph in terms of the largest eigenvalue of its adjacency matrix. We also present another upper bound using the largest eigenvalue and the size of the graph. Our bounds are asymptotically tight for some infinite families of graphs and provide improvements on the known bounds for the Grundy number of sparse random graphs.
图形的格兰迪数是指在不考虑初始顶点排序的情况下，使用第一拟合贪婪算法为图形正确着色所需的最少颜色数。计算图形的格兰迪数是一个 NP-Hard（近乎困难）问题。从诱导子图的角度来看，有这样一个特征：当且仅当一个图包含一个 k 原子时，该图的格兰迪数至少为 k。在本文中，我们利用匹配多项式的特性，确定了 k 原子的最小最大特征值。有了这一结果，我们根据图的邻接矩阵的最大特征值提出了图的格兰迪数上限。我们还利用最大特征值和图的大小提出了另一个上限。对于某些无限图族，我们的上界是渐近紧密的，并且改进了稀疏随机图的格兰迪数的已知上界。
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.
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