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Defective Coloring is Perfect for Minors 瑕疵着色非常适合未成年人
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-02-21 DOI: 10.1007/s00493-024-00081-8
Chun-Hung Liu
{"title":"Defective Coloring is Perfect for Minors","authors":"Chun-Hung Liu","doi":"10.1007/s00493-024-00081-8","DOIUrl":"https://doi.org/10.1007/s00493-024-00081-8","url":null,"abstract":"<p>The defective chromatic number of a graph class is the infimum <i>k</i> such that there exists an integer <i>d</i> such that every graph in this class can be partitioned into at most <i>k</i> induced subgraphs with maximum degree at most <i>d</i>. Finding the defective chromatic number is a fundamental graph partitioning problem and received attention recently partially due to Hadwiger’s conjecture about coloring minor-closed families. In this paper, we prove that the defective chromatic number of any minor-closed family equals the simple lower bound obtained by the standard construction, confirming a conjecture of Ossona de Mendez, Oum, and Wood. This result provides the optimal list of unavoidable finite minors for infinite graphs that cannot be partitioned into a fixed finite number of induced subgraphs with uniformly bounded maximum degree. As corollaries about clustered coloring, we obtain a linear relation between the clustered chromatic number of any minor-closed family and the tree-depth of its forbidden minors, improving an earlier exponential bound proved by Norin, Scott, Seymour, and Wood and confirming the planar case of their conjecture.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"244 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139938771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An Upper Bound for the Height of a Tree with a Given Eigenvalue 给定特征值的树高上限
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-02-02 DOI: 10.1007/s00493-023-00071-2
Artūras Dubickas
{"title":"An Upper Bound for the Height of a Tree with a Given Eigenvalue","authors":"Artūras Dubickas","doi":"10.1007/s00493-023-00071-2","DOIUrl":"https://doi.org/10.1007/s00493-023-00071-2","url":null,"abstract":"<p>In this paper we prove that every totally real algebraic integer <span>(lambda )</span> of degree <span>(d ge 2)</span> occurs as an eigenvalue of some tree of height at most <span>(d(d+1)/2+3)</span>. In order to prove this, for a given algebraic number <span>(alpha ne 0)</span>, we investigate an additive semigroup that contains zero and is closed under the map <span>(x mapsto alpha /(1-x))</span> for <span>(x ne 1)</span>. The problem of finding the smallest such semigroup seems to be of independent interest.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"26 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139660300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Generating Rank and Embedding Rank of the Hexagonic Lie Incidence Geometries 论六方列入射几何的生成秩和嵌入秩
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-01-05 DOI: 10.1007/s00493-023-00075-y
{"title":"On the Generating Rank and Embedding Rank of the Hexagonic Lie Incidence Geometries","authors":"","doi":"10.1007/s00493-023-00075-y","DOIUrl":"https://doi.org/10.1007/s00493-023-00075-y","url":null,"abstract":"<h3>Abstract</h3> <p>Given a (thick) irreducible spherical building <span> <span>(Omega )</span> </span>, we establish a bound on the difference between the generating rank and the embedding rank of its long root geometry and the dimension of the corresponding Weyl module, by showing that this difference does not grow when taking certain residues of <span> <span>(Omega )</span> </span> (in particular the residue of a vertex corresponding to a point of the long root geometry, but also other types of vertices occur). We apply this to the finite case to obtain new results on the generating rank of mainly the exceptional long root geometries, answering an open question by Cooperstein about the generating ranks of the exceptional long root subgroup geometries. We completely settle the finite case for long root geometries of type <span> <span>({{textsf{A}}}_n)</span> </span>, and the case of type <span> <span>(mathsf {F_{4,4}})</span> </span> over any field with characteristic distinct from 2 (which is not a long root subgroup geometry, but a hexagonic geometry).</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"101 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139110255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Euler’s Theorem for Regular CW-Complexes 正规 CW 复数的欧拉定理
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-01-05 DOI: 10.1007/s00493-023-00080-1
Richard H. Hammack, Paul C. Kainen
{"title":"Euler’s Theorem for Regular CW-Complexes","authors":"Richard H. Hammack, Paul C. Kainen","doi":"10.1007/s00493-023-00080-1","DOIUrl":"https://doi.org/10.1007/s00493-023-00080-1","url":null,"abstract":"<p>For strongly connected, pure <i>n</i>-dimensional regular CW-complexes, we show that <i>evenness</i> (each <span>((n{-}1))</span>-cell is contained in an even number of <i>n</i>-cells) is equivalent to generalizations of both cycle decomposition and traversability.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"48 9 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139110246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Tight Bound on Treedepth in Terms of Pathwidth and Longest Path 用路径宽度和最长路径严格约束树丛深度
IF 1.1 2区 数学
Combinatorica Pub Date : 2023-12-19 DOI: 10.1007/s00493-023-00077-w
{"title":"Tight Bound on Treedepth in Terms of Pathwidth and Longest Path","authors":"","doi":"10.1007/s00493-023-00077-w","DOIUrl":"https://doi.org/10.1007/s00493-023-00077-w","url":null,"abstract":"<h3>Abstract</h3> <p>We show that every graph with pathwidth strictly less than <em>a</em> that contains no path on <span> <span>(2^b)</span> </span> vertices as a subgraph has treedepth at most 10<em>ab</em>. The bound is best possible up to a constant factor.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"34 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138740534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Edge-Connectivity and Pairwise Disjoint Perfect Matchings in Regular Graphs 正则图中的边连接性和成对不相交完全匹配
IF 1.1 2区 数学
Combinatorica Pub Date : 2023-12-19 DOI: 10.1007/s00493-023-00078-9
Yulai Ma, Davide Mattiolo, Eckhard Steffen, Isaak H. Wolf
{"title":"Edge-Connectivity and Pairwise Disjoint Perfect Matchings in Regular Graphs","authors":"Yulai Ma, Davide Mattiolo, Eckhard Steffen, Isaak H. Wolf","doi":"10.1007/s00493-023-00078-9","DOIUrl":"https://doi.org/10.1007/s00493-023-00078-9","url":null,"abstract":"<p>For <span>(0 le t le r)</span> let <i>m</i>(<i>t</i>, <i>r</i>) be the maximum number <i>s</i> such that every <i>t</i>-edge-connected <i>r</i>-graph has <i>s</i> pairwise disjoint perfect matchings. There are only a few values of <i>m</i>(<i>t</i>, <i>r</i>) known, for instance <span>(m(3,3)=m(4,r)=1)</span>, and <span>(m(t,r) le r-2)</span> for all <span>(t not = 5)</span>, and <span>(m(t,r) le r-3)</span> if <i>r</i> is even. We prove that <span>(m(2l,r) le 3l - 6)</span> for every <span>(l ge 3)</span> and <span>(r ge 2 l)</span>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"38 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138740588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Topological Version of Hedetniemi’s Conjecture for Equivariant Spaces 赫德涅米等价空间猜想的拓扑版本
IF 1.1 2区 数学
Combinatorica Pub Date : 2023-12-19 DOI: 10.1007/s00493-023-00079-8
Vuong Bui, Hamid Reza Daneshpajouh
{"title":"A Topological Version of Hedetniemi’s Conjecture for Equivariant Spaces","authors":"Vuong Bui, Hamid Reza Daneshpajouh","doi":"10.1007/s00493-023-00079-8","DOIUrl":"https://doi.org/10.1007/s00493-023-00079-8","url":null,"abstract":"<p>A topological version of the famous Hedetniemi conjecture says: The mapping index of the Cartesian product of two <span>({mathbb {Z}}/2)</span>- spaces is equal to the minimum of their <span>({mathbb {Z}}/2)</span>-indexes. The main purpose of this article is to study the topological version of the Hedetniemi conjecture for <i>G</i>-spaces. Indeed, we show that the topological Hedetniemi conjecture cannot be valid for general pairs of <i>G</i>-spaces. More precisely, we show that this conjecture can possibly survive if the group <i>G</i> is either a cyclic <i>p</i>-group or a generalized quaternion group whose size is a power of 2.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"16 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138740501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Group Ring Approach to Fuglede’s Conjecture in Cyclic Groups 循环群中Fuglede猜想的群环逼近
IF 1.1 2区 数学
Combinatorica Pub Date : 2023-11-27 DOI: 10.1007/s00493-023-00076-x
Tao Zhang
{"title":"A Group Ring Approach to Fuglede’s Conjecture in Cyclic Groups","authors":"Tao Zhang","doi":"10.1007/s00493-023-00076-x","DOIUrl":"https://doi.org/10.1007/s00493-023-00076-x","url":null,"abstract":"<p>Fuglede’s conjecture states that a subset <span>(Omega subseteq mathbb {R}^{n})</span> with positive and finite Lebesgue measure is a spectral set if and only if it tiles <span>(mathbb {R}^{n})</span> by translation. However, this conjecture does not hold in both directions for <span>(mathbb {R}^n)</span>, <span>(nge 3)</span>. While the conjecture remains unsolved in <span>(mathbb {R})</span> and <span>(mathbb {R}^2)</span>, cyclic groups are instrumental in its study within <span>(mathbb {R})</span>. This paper introduces a new tool to study spectral sets in cyclic groups and, in particular, proves that Fuglede’s conjecture holds in <span>(mathbb {Z}_{p^{n}qr})</span>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"92 22","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138442741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
A Characterization of Graphs Whose Small Powers of Their Edge Ideals Have a Linear Free Resolution 图的小幂边理想具有线性自由分辨率的刻画
IF 1.1 2区 数学
Combinatorica Pub Date : 2023-11-27 DOI: 10.1007/s00493-023-00074-z
Nguyen Cong Minh, Thanh Vu
{"title":"A Characterization of Graphs Whose Small Powers of Their Edge Ideals Have a Linear Free Resolution","authors":"Nguyen Cong Minh, Thanh Vu","doi":"10.1007/s00493-023-00074-z","DOIUrl":"https://doi.org/10.1007/s00493-023-00074-z","url":null,"abstract":"<p>Let <i>I</i>(<i>G</i>) be the edge ideal of a simple graph <i>G</i>. We prove that <span>(I(G)^2)</span> has a linear free resolution if and only if <i>G</i> is gap-free and <span>({{,textrm{reg},}}I(G) le 3)</span>. Similarly, we show that <span>(I(G)^3)</span> has a linear free resolution if and only if <i>G</i> is gap-free and <span>({{,textrm{reg},}}I(G) le 4)</span>. We deduce these characterizations by establishing a general formula for the regularity of powers of edge ideals of gap-free graphs <span>({{,textrm{reg},}}(I(G)^s) = max ({{,textrm{reg},}}I(G) + s-1,2s))</span>, for <span>(s =2,3)</span>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"97 30","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138442303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Universal Planar Graphs for the Topological Minor Relation 拓扑小关系的通用平面图
IF 1.1 2区 数学
Combinatorica Pub Date : 2023-11-21 DOI: 10.1007/s00493-023-00073-0
Florian Lehner
{"title":"Universal Planar Graphs for the Topological Minor Relation","authors":"Florian Lehner","doi":"10.1007/s00493-023-00073-0","DOIUrl":"https://doi.org/10.1007/s00493-023-00073-0","url":null,"abstract":"<p>Huynh et al. recently showed that a countable graph <i>G</i> which contains every countable planar graph as a subgraph must contain arbitrarily large finite complete graphs as topological minors, and an infinite complete graph as a minor. We strengthen this result by showing that the same conclusion holds if <i>G</i> contains every countable planar graph as a topological minor. In particular, there is no countable planar graph containing every countable planar graph as a topological minor, answering a question by Diestel and Kühn. Moreover, we construct a locally finite planar graph which contains every locally finite planar graph as a topological minor. This shows that in the above result it is not enough to require that <i>G</i> contains every locally finite planar graph as a topological minor.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"27 19","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138293813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
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