Growth rates of the bipartite Erdős–Gyárfás function

IF 0.9 3区 数学 Q2 MATHEMATICS
Xihe Li, Hajo Broersma, Ligong Wang
{"title":"Growth rates of the bipartite Erdős–Gyárfás function","authors":"Xihe Li,&nbsp;Hajo Broersma,&nbsp;Ligong Wang","doi":"10.1002/jgt.23149","DOIUrl":null,"url":null,"abstract":"<p>Given two graphs <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n \n <mo>,</mo>\n \n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> $G,H$</annotation>\n </semantics></math> and a positive integer <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>q</mi>\n </mrow>\n </mrow>\n <annotation> $q$</annotation>\n </semantics></math>, an <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>H</mi>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $(H,q)$</annotation>\n </semantics></math>-coloring of <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is an edge-coloring of <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> such that every copy of <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math> in <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> receives at least <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>q</mi>\n </mrow>\n </mrow>\n <annotation> $q$</annotation>\n </semantics></math> distinct colors. The bipartite Erdős–Gyárfás function <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>n</mi>\n </mrow>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>K</mi>\n \n <mrow>\n <mi>s</mi>\n \n <mo>,</mo>\n \n <mi>t</mi>\n </mrow>\n </msub>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $r({K}_{n,n},{K}_{s,t},q)$</annotation>\n </semantics></math> is defined to be the minimum number of colors needed for <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>n</mi>\n </mrow>\n </msub>\n </mrow>\n </mrow>\n <annotation> ${K}_{n,n}$</annotation>\n </semantics></math> to have a <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mrow>\n <mi>s</mi>\n \n <mo>,</mo>\n \n <mi>t</mi>\n </mrow>\n </msub>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $({K}_{s,t},q)$</annotation>\n </semantics></math>-coloring. For balanced complete bipartite graphs <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mrow>\n <mi>p</mi>\n \n <mo>,</mo>\n \n <mi>p</mi>\n </mrow>\n </msub>\n </mrow>\n </mrow>\n <annotation> ${K}_{p,p}$</annotation>\n </semantics></math>, the function <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>n</mi>\n </mrow>\n </msub>\n \n <mo>,</mo>\n \n <msub>\n <mi>K</mi>\n \n <mrow>\n <mi>p</mi>\n \n <mo>,</mo>\n \n <mi>p</mi>\n </mrow>\n </msub>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $r({K}_{n,n},{K}_{p,p},q)$</annotation>\n </semantics></math> was studied systematically in Axenovich et al. In this paper, we study the asymptotic behavior of this function for complete bipartite graphs <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mrow>\n <mi>s</mi>\n \n <mo>,</mo>\n \n <mi>t</mi>\n </mrow>\n </msub>\n </mrow>\n </mrow>\n <annotation> ${K}_{s,t}$</annotation>\n </semantics></math> that are not necessarily balanced. Our main results deal with thresholds and lower and upper bounds for the growth rate of this function, in particular for (sub)linear and (sub)quadratic growth. We also obtain new lower bounds for the balanced bipartite case, and improve several results given by Axenovich, Füredi and Mubayi. Our proof techniques are based on an extension to bipartite graphs of the recently developed Color Energy Method by Pohoata and Sheffer and its refinements, and a generalization of an old result due to Corrádi.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"597-628"},"PeriodicalIF":0.9000,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23149","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23149","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

Given two graphs G , H $G,H$ and a positive integer q $q$ , an ( H , q ) $(H,q)$ -coloring of G $G$ is an edge-coloring of G $G$ such that every copy of H $H$ in G $G$ receives at least q $q$ distinct colors. The bipartite Erdős–Gyárfás function r ( K n , n , K s , t , q ) $r({K}_{n,n},{K}_{s,t},q)$ is defined to be the minimum number of colors needed for K n , n ${K}_{n,n}$ to have a ( K s , t , q ) $({K}_{s,t},q)$ -coloring. For balanced complete bipartite graphs K p , p ${K}_{p,p}$ , the function r ( K n , n , K p , p , q ) $r({K}_{n,n},{K}_{p,p},q)$ was studied systematically in Axenovich et al. In this paper, we study the asymptotic behavior of this function for complete bipartite graphs K s , t ${K}_{s,t}$ that are not necessarily balanced. Our main results deal with thresholds and lower and upper bounds for the growth rate of this function, in particular for (sub)linear and (sub)quadratic growth. We also obtain new lower bounds for the balanced bipartite case, and improve several results given by Axenovich, Füredi and Mubayi. Our proof techniques are based on an extension to bipartite graphs of the recently developed Color Energy Method by Pohoata and Sheffer and its refinements, and a generalization of an old result due to Corrádi.

Abstract Image

双向厄尔多斯-京法斯函数的增长率
给定两个图和一个正整数 , 的-着色是一个边着色,使得 in 的每一个副本都得到至少不同的颜色。双向厄尔多斯-雅尔法函数的定义是,要有一个-着色所需的最少颜色数。在本文中,我们将研究该函数在不一定平衡的完整双方图中的渐近行为。我们的主要结果涉及该函数增长率的阈值、下限和上限,特别是(亚)线性增长和(亚)二次增长。我们还获得了平衡两方情况下的新下限,并改进了阿克森诺维奇、富雷迪和穆巴伊给出的几个结果。我们的证明技术是基于 Pohoata 和 Sheffer 最近开发的彩色能量法及其改进版,以及对 Corrádi 提出的一个旧结果的推广。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Journal of Graph Theory
Journal of Graph Theory 数学-数学
CiteScore
1.60
自引率
22.20%
发文量
130
审稿时长
6-12 weeks
期刊介绍: The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信