{"title":"Finding irregular subgraphs via local adjustments","authors":"Jie Ma , 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, 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}
Xiying Du , António Girão , Zach Hunter , Rose McCarty , Alex Scott
{"title":"Induced C4-free subgraphs with large average degree","authors":"Xiying Du , António Girão , Zach Hunter , Rose McCarty , 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 , 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 , 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}
{"title":"Pivot-minors and the Erdős-Hajnal conjecture","authors":"James Davies","doi":"10.1016/j.jctb.2025.04.004","DOIUrl":"10.1016/j.jctb.2025.04.004","url":null,"abstract":"<div><div>We prove a conjecture of Kim and Oum that every proper pivot-minor-closed class of graphs has the strong Erdős-Hajnal property. More precisely, for every graph <em>H</em>, there exists <span><math><mi>ϵ</mi><mo>></mo><mn>0</mn></math></span> such that every <em>n</em>-vertex graph with no pivot-minor isomorphic to <em>H</em> contains two sets <span><math><mi>A</mi><mo>,</mo><mi>B</mi></math></span> of vertices such that <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>,</mo><mo>|</mo><mi>B</mi><mo>|</mo><mo>⩾</mo><mi>ϵ</mi><mi>n</mi></math></span> and <em>A</em> is complete or anticomplete to <em>B</em>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 257-278"},"PeriodicalIF":1.2,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143834331","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}
Rutger Campbell , J. Pascal Gollin , Kevin Hendrey , Raphael Steiner
{"title":"Optimal bounds for zero-sum cycles. I. Odd order","authors":"Rutger Campbell , J. Pascal Gollin , Kevin Hendrey , Raphael Steiner","doi":"10.1016/j.jctb.2025.04.003","DOIUrl":"10.1016/j.jctb.2025.04.003","url":null,"abstract":"<div><div>For a finite (not necessarily abelian) group <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><mo>⋅</mo><mo>)</mo></math></span>, let <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> denote the smallest positive integer <em>n</em> such that for each labelling of the arcs of the complete digraph of order <em>n</em> using elements from Γ, there exists a directed cycle such that the arc-labels along the cycle multiply to the identity. Alon and Krivelevich <span><span>[2]</span></span> initiated the study of the parameter <span><math><mi>n</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math></span> on cyclic groups and proved <span><math><mi>n</mi><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><mi>q</mi><mi>log</mi><mo></mo><mi>q</mi><mo>)</mo></math></span>. This was later improved to a linear bound of <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mn>8</mn><mo>|</mo><mi>Γ</mi><mo>|</mo></math></span> for every finite abelian group by Mészáros and the last author <span><span>[8]</span></span>, and then further to <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mo>|</mo><mi>Γ</mi><mo>|</mo><mo>−</mo><mn>1</mn></math></span> for every non-trivial finite group independently by Berendsohn, Boyadzhiyska and Kozma <span><span>[3]</span></span> as well as by Akrami, Alon, Chaudhury, Garg, Mehlhorn and Mehta <span><span>[1]</span></span>.</div><div>In this series of two papers we conclude this line of research by proving that <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mo>|</mo><mi>Γ</mi><mo>|</mo><mo>+</mo><mn>1</mn></math></span> for every finite group <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><mo>⋅</mo><mo>)</mo></math></span>, which is the best possible such bound in terms of the group order and precisely determines the value of <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> for all cyclic groups as <span><math><mi>n</mi><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>q</mi><mo>+</mo><mn>1</mn></math></span>.</div><div>In the present paper we prove the above result for all groups of odd order. The proof for groups of even order needs to overcome substantial additional obstacles and will be presented in the second part of this series.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 246-256"},"PeriodicalIF":1.2,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143834330","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":"Embedding connected factorizations II","authors":"Amin Bahmanian , Anna Johnsen-Yu","doi":"10.1016/j.jctb.2025.03.003","DOIUrl":"10.1016/j.jctb.2025.03.003","url":null,"abstract":"<div><div>Let <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> be the complete <em>h</em>-uniform <em>n</em>-vertex hypergraph in which each edge is repeated <em>λ</em> times. For <span><math><mi>r</mi><mo>:</mo><mo>=</mo><mo>(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, a <em>(partial)</em> <strong>r</strong><em>-factorization</em> of <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> is a partition of the edges of <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> into factors <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> such that each factor is spanning and the degree of all vertices in each <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is (at most) <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. Suppose that <span><math><mi>n</mi><mo>≥</mo><mo>(</mo><mi>h</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>. We establish necessary and sufficient conditions that ensure a partial <strong>r</strong>-factorization of <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> can be embedded in a connected <strong>r</strong>-factorization of <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span>. Moreover, we prove a general result which leads to a complete characterization of partial <span><math><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span>-factorizations of <em>any</em> sub-hypergraph of <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> in connected <strong>r</strong>-factorizations of <span><math><mi>λ</mi><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> so long as <em>q</em> meets a natural upper bound. These results can be seen as unified generalizations of many classical combinatorial results, and can also be restated as results on embedding partial symmetric latin hypercubes.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 374-398"},"PeriodicalIF":1.2,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143894863","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 splitter theorem on 3-connected binary matroids and inner fans","authors":"João Paulo Costalonga","doi":"10.1016/j.jctb.2025.03.004","DOIUrl":"10.1016/j.jctb.2025.03.004","url":null,"abstract":"<div><div>We establish a splitter type theorem for 3-connected binary matroids regarding elements whose contraction preserves a fixed 3-connected minor and the vertical 3-connectivity. We established that, for 3-connected simple binary matroids <span><math><mi>N</mi><mo><</mo><mi>M</mi></math></span>, there is a disjoint family <span><math><mo>{</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo><mo>⊆</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>E</mi><mo>(</mo><mi>M</mi><mo>)</mo></mrow></msup></math></span> such that <span><math><mi>r</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>r</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>r</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><mo>⋯</mo><mo>∪</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>≥</mo><mi>r</mi><mo>(</mo><mi>M</mi><mo>)</mo><mo>−</mo><mi>r</mi><mo>(</mo><mi>N</mi><mo>)</mo></math></span>, each <span><math><mrow><mi>si</mi></mrow><mo>(</mo><mi>M</mi><mo>/</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> is 3-connected with an <em>N</em>-minor, and either <span><math><mo>|</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>=</mo><mn>1</mn></math></span> or <em>X</em> is a special type of fan. We also establish a stronger version of this result under specific hypotheses. These results have several consequences, including the generalizations for binary matroids of some results about contractible edges in 3-connected graphs and some other structural results for graphs and binary matroids.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 204-245"},"PeriodicalIF":1.2,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739232","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}
Micha Christoph, Charlotte Knierim, Anders Martinsson, Raphael Steiner
{"title":"Improved bounds for zero-sum cycles in Zpd","authors":"Micha Christoph, Charlotte Knierim, Anders Martinsson, Raphael Steiner","doi":"10.1016/j.jctb.2025.03.001","DOIUrl":"10.1016/j.jctb.2025.03.001","url":null,"abstract":"<div><div>For a finite abelian group <span><math><mo>(</mo><mi>Γ</mi><mo>,</mo><mo>+</mo><mo>)</mo></math></span>, let <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> denote the smallest positive integer <em>n</em> such that for each labeling of the arcs of the complete digraph of order <em>n</em> using elements from Γ, there exists a directed cycle such that the total sum of the arc-labels along the cycle equals 0. Alon and Krivelevich initiated the study of the parameter <span><math><mi>n</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math></span> on cyclic groups and proved that <span><math><mi>n</mi><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><mi>q</mi><mi>log</mi><mo></mo><mi>q</mi><mo>)</mo></math></span>. Several improvements and generalizations of this bound have since been obtained, and an optimal bound in terms of the group order of the form <span><math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mo>|</mo><mi>Γ</mi><mo>|</mo><mo>+</mo><mn>1</mn></math></span> was recently announced by Campbell, Gollin, Hendrey and the last author. While this bound is tight when the group Γ is cyclic, in cases when Γ is far from being cyclic, significant improvements on the bound can be made. In this direction, studying the prototypical case when <span><math><mi>Γ</mi><mo>=</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> is a power of a cyclic group of prime order, Letzter and Morrison [<em>Journal of Combinatorial Theory Series B, 2024</em>] showed that <span><math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>O</mi><mo>(</mo><mi>p</mi><mi>d</mi><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>d</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> and that <span><math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>O</mi><mo>(</mo><mi>d</mi><mi>log</mi><mo></mo><mi>d</mi><mo>)</mo></math></span>. They then posed the problem of proving an (asymptotically optimal) upper bound of <span><math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>O</mi><mo>(</mo><mi>p</mi><mi>d</mi><mo>)</mo></math></span> for all primes <em>p</em> and <span><math><mi>d</mi><mo>∈</mo><mi>N</mi></math></span>. In this paper, we solve this problem for <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> and improve their bound for all primes <span><math><mi>p</mi><mo>≥</mo><mn>3</mn></math></span> by proving <span><math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mn>5</mn><mi>d</mi></math></span> and <span><math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 365-373"},"PeriodicalIF":1.2,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143894864","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}
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