Journal of Combinatorial Theory Series B最新文献

{"title":"On the Keevash-Knox-Mycroft conjecture","authors":"Luyining Gan ,&nbsp;Jie Han","doi":"10.1016/j.jctb.2025.05.003","DOIUrl":"10.1016/j.jctb.2025.05.003","url":null,"abstract":"<div><div>Given <span><math><mn>1</mn><mo>≤</mo><mi>ℓ</mi><mo>&lt;</mo><mi>k</mi></math></span> and <span><math><mi>δ</mi><mo>≥</mo><mn>0</mn></math></span>, let <span><math><mtext>PM</mtext><mo>(</mo><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span> be the decision problem for the existence of perfect matchings in <em>n</em>-vertex <em>k</em>-uniform hypergraphs with minimum <em>ℓ</em>-degree at least <span><math><mi>δ</mi><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>ℓ</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mi>ℓ</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. For <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, <span><math><mtext>PM</mtext><mo>(</mo><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span> was one of the first NP-complete problems by Karp. Keevash, Knox and Mycroft conjectured that <span><math><mtext>PM</mtext><mo>(</mo><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span> is in P for every <span><math><mi>δ</mi><mo>&gt;</mo><mn>1</mn><mo>−</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>k</mi><mo>)</mo></mrow><mrow><mi>k</mi><mo>−</mo><mi>ℓ</mi></mrow></msup></math></span> and verified the case <span><math><mi>ℓ</mi><mo>=</mo><mi>k</mi><mo>−</mo><mn>1</mn></math></span>.</div><div>In this paper we show that this problem can be reduced to the study of the minimum <em>ℓ</em>-degree condition forcing the existence of fractional perfect matchings. Together with existing results on fractional perfect matchings, this solves the conjecture of Keevash, Knox and Mycroft for <span><math><mi>ℓ</mi><mo>≥</mo><mn>0.4</mn><mi>k</mi></math></span>. Moreover, we also supply an algorithm that outputs a perfect matching, provided that one exists.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"174 ","pages":"Pages 214-242"},"PeriodicalIF":1.2,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144154896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The 1-2 conjecture holds for regular graphs","authors":"Kecai Deng ,&nbsp;Hongyuan Qiu","doi":"10.1016/j.jctb.2025.05.002","DOIUrl":"10.1016/j.jctb.2025.05.002","url":null,"abstract":"<div><div>The 1-2 conjecture asserts that the vertices and edges of every graph can be assigned with weights in <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></math></span> such that adjacent vertices receive distinct weighted degrees. While this conjecture remains open in general, it has been proven that it is possible to achieve this using the weight set <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></math></span>. We demonstrate that the weight set <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span> suffices for every graph. As a corollary, the 1-2 conjecture is confirmed for regular graphs. Additionally, we verify another related conjecture concerning locally irregular total colouring, for regular graphs.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"174 ","pages":"Pages 207-213"},"PeriodicalIF":1.2,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144084655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Connectivity keeping paths containing prescribed vertices in highly connected triangle-free graphs","authors":"Shinya Fujita","doi":"10.1016/j.jctb.2025.05.001","DOIUrl":"10.1016/j.jctb.2025.05.001","url":null,"abstract":"<div><div>Let <span><math><mi>m</mi><mo>,</mo><mi>k</mi></math></span> be integers with <span><math><mi>m</mi><mo>≥</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>. For a <em>k</em>-connected graph <em>G</em>, a subgraph <em>H</em> of <em>G</em> is <em>k-removable</em> if <span><math><mi>G</mi><mo>−</mo><mi>V</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> is still a <em>k</em>-connected graph. A graph is <em>triangle-free</em> if it contains no triangle as a subgraph.</div><div>In this paper, we prove that if <em>G</em> is a <em>k</em>-connected triangle-free graph with minimum degree at least <span><math><mi>k</mi><mo>+</mo><mo>(</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span>, then for any vertex <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, there exists a path <em>P</em> on <em>m</em> vertices starting from <em>v</em> such that <span><math><mi>G</mi><mo>−</mo><mi>V</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> is a <span><math><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-connected graph. This result is obtained by showing a stronger statement concerning the existence of <em>k</em>-removable paths in <em>k</em>-connected triangle-free graphs. We also prove that if <em>G</em> is a <em>k</em>-connected triangle-free graph with minimum degree at least <span><math><mi>k</mi><mo>+</mo><mn>1</mn></math></span>, then <em>G</em> contains a <em>k</em>-removable edge. Our results confirm a conjecture due to Luo et al. concerning the existence of a <em>k</em>-removable path on <em>m</em> vertices in a <em>k</em>-connected bipartite graph for all odd <em>m</em> together with the case <span><math><mi>m</mi><mo>=</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"174 ","pages":"Pages 190-206"},"PeriodicalIF":1.2,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144068351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A structure theorem for pseudosegments and its applications","authors":"Jacob Fox ,&nbsp;János Pach ,&nbsp;Andrew Suk","doi":"10.1016/j.jctb.2025.04.007","DOIUrl":"10.1016/j.jctb.2025.04.007","url":null,"abstract":"<div><div>We prove a far-reaching strengthening of Szemerédi's regularity lemma for intersection graphs of pseudosegments. It shows that the vertex set of such a graph can be partitioned into a bounded number of parts of roughly the same size such that almost all bipartite graphs between different pairs of parts are <em>complete</em> or <em>empty</em>. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every <em>n</em>-vertex simple topological graph with no <em>k</em> pairwise disjoint edges has at most <span><math><mi>n</mi><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> edges.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"174 ","pages":"Pages 99-132"},"PeriodicalIF":1.2,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143905987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A characterization of testable hypergraph properties","authors":"Felix Joos ,&nbsp;Jaehoon Kim ,&nbsp;Daniela Kühn ,&nbsp;Deryk Osthus","doi":"10.1016/j.jctb.2025.04.009","DOIUrl":"10.1016/j.jctb.2025.04.009","url":null,"abstract":"<div><div>We provide a combinatorial characterization of all testable properties of <em>k</em>-uniform hypergraphs (<em>k</em>-graphs for short). Here, a <em>k</em>-graph property <strong>P</strong> is testable if there is a randomized algorithm which makes a bounded number of edge queries and distinguishes with probability 2/3 between <em>k</em>-graphs that satisfy <strong>P</strong> and those that are far from satisfying <strong>P</strong>. For the 2-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. Our results for the <em>k</em>-graph setting are in contrast to those of Austin and Tao, who showed that for the somewhat stronger concept of local repairability, the testability results for graphs do not extend to the 3-graph setting.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"174 ","pages":"Pages 133-189"},"PeriodicalIF":1.2,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143912685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Finding irregular subgraphs via local adjustments","authors":"Jie Ma ,&nbsp;Shengjie Xie","doi":"10.1016/j.jctb.2025.04.008","DOIUrl":"10.1016/j.jctb.2025.04.008","url":null,"abstract":"<div><div>For a graph <em>H</em>, let <span><math><mi>m</mi><mo>(</mo><mi>H</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> denote the number of vertices of degree <em>k</em> in <em>H</em>. A conjecture of Alon and Wei states that for any <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>, every <em>n</em>-vertex <em>d</em>-regular graph contains a spanning subgraph <em>H</em> satisfying <span><math><mo>|</mo><mi>m</mi><mo>(</mo><mi>H</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>−</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>|</mo><mo>≤</mo><mn>2</mn></math></span> for every <span><math><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>d</mi></math></span>. This holds easily when <span><math><mi>d</mi><mo>≤</mo><mn>2</mn></math></span>. An asymptotic version of this conjecture was initially established by Frieze, Gould, Karoński and Pfender, subsequently improved by Alon and Wei, and most recently enhanced by Fox, Luo and Pham, approaching its complete range. All of these approaches relied on probabilistic methods.</div><div>In this paper, we provide a novel framework to study this conjecture, based on localized deterministic techniques which we call local adjustments. We prove two main results. Firstly, we show that every <em>n</em>-vertex <em>d</em>-regular graph contains a spanning subgraph <em>H</em> satisfying <span><math><mo>|</mo><mi>m</mi><mo>(</mo><mi>H</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>−</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>|</mo><mo>≤</mo><mn>2</mn><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for all <span><math><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>d</mi></math></span>, which provides the first bound independent of the value of <em>n</em>. Secondly, we confirm the case <span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span> of the Alon-Wei Conjecture in a strong form. Both results can be generalized to multigraphs and yield efficient algorithms for finding the desired subgraphs <em>H</em>. Furthermore, we explore a generalization of the Alon-Wei Conjecture for multigraphs and its connection to the Faudree-Lehel Conjecture concerning irregularity strength.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"174 ","pages":"Pages 71-98"},"PeriodicalIF":1.2,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143895437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weak diameter choosability of graphs with an excluded minor","authors":"Joshua Crouch,&nbsp;Chun-Hung Liu","doi":"10.1016/j.jctb.2025.04.005","DOIUrl":"10.1016/j.jctb.2025.04.005","url":null,"abstract":"<div><div>Weak diameter coloring of graphs recently attracted attention, partially due to its connection to asymptotic dimension of metric spaces. We consider weak diameter list-coloring of graphs in this paper. Dvořák and Norin proved that graphs with bounded Euler genus are 3-choosable with bounded weak diameter. In this paper, we extend their result by showing that for every graph <em>H</em>, <em>H</em>-minor free graphs are 3-choosable with bounded weak diameter. The upper bound 3 is optimal and it strengthens an earlier result for non-list-coloring <em>H</em>-minor free graphs with bounded weak diameter. As a corollary, <em>H</em>-minor free graphs with bounded maximum degree are 3-choosable with bounded clustering, strengthening an earlier result for non-list-coloring.</div><div>When <em>H</em> is planar, we prove a much stronger result: for every 2-list-assignment <em>L</em> of an <em>H</em>-minor free graph, every precoloring with bounded weak diameter can be extended to an <em>L</em>-coloring with bounded weak diameter. It is a common generalization of earlier results for non-list-coloring with bounded weak diameter and for list-coloring with bounded clustering without allowing precolorings. As a corollary, for any planar graph <em>H</em> and <em>H</em>-minor free graph <em>G</em>, there are exponentially many list-colorings of <em>G</em> with bounded weak diameter (and with bounded clustering if <em>G</em> also has bounded maximum degree); and every graph with bounded layered tree-width and bounded maximum degree has exponentially many 3-colorings with bounded clustering.</div><div>We also show that the aforementioned results for list-coloring cannot be extended to odd minor free graphs by showing that some bipartite graphs with maximum degree Δ are <em>k</em>-choosable with bounded weak diameter only when <span><math><mi>k</mi><mo>=</mo><mi>Ω</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>Δ</mi><mo>/</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>Δ</mi><mo>)</mo></math></span>. On the other hand, we show that odd <em>H</em>-minor graphs are 3-colorable with bounded weak diameter, implying an earlier result about clustered coloring of odd <em>H</em>-minor free graphs with bounded maximum degree.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"174 ","pages":"Pages 28-70"},"PeriodicalIF":1.2,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143864224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Induced C4-free subgraphs with large average degree","authors":"Xiying Du ,&nbsp;António Girão ,&nbsp;Zach Hunter ,&nbsp;Rose McCarty ,&nbsp;Alex Scott","doi":"10.1016/j.jctb.2025.04.002","DOIUrl":"10.1016/j.jctb.2025.04.002","url":null,"abstract":"<div><div>We prove that there exists a constant <em>C</em> so that, for all <span><math><mi>s</mi><mo>,</mo><mi>k</mi><mo>∈</mo><mi>N</mi></math></span>, if <em>G</em> has average degree at least <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>C</mi><msup><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msup></math></span> and does not contain <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>s</mi></mrow></msub></math></span> as a subgraph then it contains an induced subgraph which is <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free and has average degree at least <em>k</em>. It was known that some function of <em>s</em> and <em>k</em> suffices, but this is the first explicit bound. We give several applications of this result, including short and streamlined proofs of the following two corollaries.</div><div>We show that there exists a constant <em>C</em> so that, for all <span><math><mi>s</mi><mo>,</mo><mi>k</mi><mo>∈</mo><mi>N</mi></math></span>, if <em>G</em> has average degree at least <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>C</mi><msup><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msup></math></span> and does not contain <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>s</mi></mrow></msub></math></span> as a subgraph then it contains an induced subdivision of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>. This is the first quantitative improvement on a well-known theorem of Kühn and Osthus; their proof gives a bound that is triply exponential in both <em>k</em> and <em>s</em>.</div><div>We also show that for any hereditary degree-bounded class <span><math><mi>F</mi></math></span>, there exists a constant <span><math><mi>C</mi><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> so that <span><math><msup><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msup></math></span> is a degree-bounding function for <span><math><mi>F</mi></math></span>. This is the first bound of any type on the rate of growth of such functions.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 305-328"},"PeriodicalIF":1.2,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143860680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A matrix realization of spectral bounds","authors":"Yen-Jen Cheng ,&nbsp;Chih-wen Weng","doi":"10.1016/j.jctb.2025.04.006","DOIUrl":"10.1016/j.jctb.2025.04.006","url":null,"abstract":"<div><div>We give a unified and systematic way to find bounds for the largest real eigenvalue of a nonnegative matrix by considering its modified quotient matrix. We leverage this insight to identify the unique matrix whose largest real eigenvalue is maximum among all <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-matrices with a specified number of ones. This result resolves a problem that was posed independently by R. Brualdi and A. Hoffman, as well as F. Friedland, back in 1985.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"174 ","pages":"Pages 1-27"},"PeriodicalIF":1.2,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143864170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Haar graphical representations of finite groups and an application to poset representations","authors":"Joy Morris ,&nbsp;Pablo Spiga","doi":"10.1016/j.jctb.2025.04.001","DOIUrl":"10.1016/j.jctb.2025.04.001","url":null,"abstract":"<div><div>Let <em>R</em> be a group and let <em>S</em> be a subset of <em>R</em>. The Haar graph <span><math><mrow><mi>Haar</mi></mrow><mo>(</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span> of <em>R</em> with connection set <em>S</em> is the graph having vertex set <span><math><mi>R</mi><mo>×</mo><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>, where two distinct vertices <span><math><mo>(</mo><mi>x</mi><mo>,</mo><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>y</mi><mo>,</mo><mn>1</mn><mo>)</mo></math></span> are declared to be adjacent if and only if <span><math><mi>y</mi><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>∈</mo><mi>S</mi></math></span>. The name Haar graph was coined by Tomaž Pisanski in one of the first investigations on this class of graphs.</div><div>For every <span><math><mi>g</mi><mo>∈</mo><mi>R</mi></math></span>, the mapping <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>:</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>ε</mi><mo>)</mo><mo>↦</mo><mo>(</mo><mi>x</mi><mi>g</mi><mo>,</mo><mi>ε</mi><mo>)</mo></math></span>, <span><math><mo>∀</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>ε</mi><mo>)</mo><mo>∈</mo><mi>R</mi><mo>×</mo><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>, is an automorphism of <span><math><mrow><mi>Haar</mi></mrow><mo>(</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span>. In particular, the set <span><math><mover><mrow><mi>R</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>:</mo><mo>=</mo><mo>{</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>|</mo><mi>g</mi><mo>∈</mo><mi>R</mi><mo>}</mo></math></span> is a subgroup of the automorphism group of <span><math><mrow><mi>Haar</mi></mrow><mo>(</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span> isomorphic to <em>R</em>. In the case that the automorphism group of <span><math><mrow><mi>Haar</mi></mrow><mo>(</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span> equals <span><math><mover><mrow><mi>R</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>, the Haar graph <span><math><mrow><mi>Haar</mi></mrow><mo>(</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span> is said to be a Haar graphical representation of the group <em>R</em>.</div><div>Answering a question of Feng, Kovács, Wang, and Yang, we classify the finite groups admitting a Haar graphical representation. Specifically, we show that every finite group admits a Haar graphical representation, with abelian groups and ten other small groups as the only exceptions.</div><div>Our work on Haar graphs allows us to improve a 1980 result of Babai concerning representations of groups on posets, achieving the best possible result in this direction. An improvement to Babai's related result on representations of groups on distributive lattices follows.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 279-304"},"PeriodicalIF":1.2,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143834332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}