Journal of Combinatorial Theory Series B最新文献

{"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":"2025-07-01","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>&gt;</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":"2025-07-01","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}

{"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":"2025-07-01","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":"Every d(d + 1)-connected graph is globally rigid in Rd","authors":"Soma Villányi","doi":"10.1016/j.jctb.2025.01.005","DOIUrl":"10.1016/j.jctb.2025.01.005","url":null,"abstract":"<div><div>Using a probabilistic method, we prove that <span><math><mi>d</mi><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-connected graphs are rigid in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, a conjecture of Lovász and Yemini. Then, using recent results on weakly globally linked pairs, we modify our argument to prove that <span><math><mi>d</mi><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-connected graphs are globally rigid, too, a conjecture of Connelly, Jordán and Whiteley. The constant <span><math><mi>d</mi><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span> is best possible.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 1-13"},"PeriodicalIF":1.2,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal bounds for zero-sum cycles. I. Odd order","authors":"Rutger Campbell ,&nbsp;J. Pascal Gollin ,&nbsp;Kevin Hendrey ,&nbsp;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":"2025-07-01","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":"Ramsey numbers of bounded degree trees versus general graphs","authors":"Richard Montgomery ,&nbsp;Matías Pavez-Signé ,&nbsp;Jun Yan","doi":"10.1016/j.jctb.2025.02.004","DOIUrl":"10.1016/j.jctb.2025.02.004","url":null,"abstract":"<div><div>For every <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and Δ, we prove that there exists a constant <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>Δ</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> such that the following holds. For every graph <em>H</em> with <span><math><mi>χ</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>=</mo><mi>k</mi></math></span> and every tree <em>T</em> with at least <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>Δ</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>|</mo><mi>H</mi><mo>|</mo></math></span> vertices and maximum degree at most Δ, the Ramsey number <span><math><mi>R</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is <span><math><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mo>|</mo><mi>T</mi><mo>|</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>σ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>, where <span><math><mi>σ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> is the size of a smallest colour class across all proper <em>k</em>-colourings of <em>H</em>. This is tight up to the value of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>Δ</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span>, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 102-145"},"PeriodicalIF":1.2,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453850","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":"Aharoni's rainbow cycle conjecture holds up to an additive constant","authors":"Patrick Hompe,&nbsp;Tony Huynh","doi":"10.1016/j.jctb.2024.12.004","DOIUrl":"10.1016/j.jctb.2024.12.004","url":null,"abstract":"<div><div>In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture: if <em>G</em> is a simple <em>n</em>-vertex edge-colored graph with <em>n</em> color classes of size at least <em>r</em>, then <em>G</em> contains a rainbow cycle of length at most <span><math><mo>⌈</mo><mi>n</mi><mo>/</mo><mi>r</mi><mo>⌉</mo></math></span>.</div><div>In this paper, we prove that, for fixed <em>r</em>, Aharoni's conjecture holds up to an additive constant. Specifically, we show that for each fixed <span><math><mi>r</mi><mo>⩾</mo><mn>1</mn></math></span>, there exists a constant <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>∈</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>r</mi></mrow><mrow><mn>5</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mi>r</mi><mo>)</mo></math></span> such that if <em>G</em> is a simple <em>n</em>-vertex edge-colored graph with <em>n</em> color classes of size at least <em>r</em>, then <em>G</em> contains a rainbow cycle of length at most <span><math><mi>n</mi><mo>/</mo><mi>r</mi><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 80-93"},"PeriodicalIF":1.2,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092865","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 saturation in graphs: A combinatorial approach","authors":"Nikolai Terekhov ,&nbsp;Maksim Zhukovskii","doi":"10.1016/j.jctb.2024.12.007","DOIUrl":"10.1016/j.jctb.2024.12.007","url":null,"abstract":"<div><div>The weak saturation number <span><math><mrow><mi>wsat</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> is the minimum number of edges in a graph on <em>n</em> vertices such that all the missing edges can be activated sequentially so that each new edge creates a copy of <em>F</em>. In contrast to previous algebraic approaches, we present a new combinatorial approach to prove lower bounds for weak saturation numbers that allows to establish worst-case tight (up to constant additive terms) general lower bounds as well as to get exact values of the weak saturation numbers for certain graph families. It is known (Alon, 1985) that, for every <em>F</em>, there exists <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> such that <span><math><mrow><mi>wsat</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>F</mi></mrow></msub><mi>n</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></math></span>. Our lower bounds imply that all values in the interval <span><math><mo>[</mo><mfrac><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>δ</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>,</mo><mi>δ</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span> with step size <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>δ</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></span> are achievable by <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> for graphs <em>F</em> with minimum degree <em>δ</em> (while any value outside this interval is not achievable).</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 146-167"},"PeriodicalIF":1.2,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143127971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Slow graph bootstrap percolation II: Accelerating properties","authors":"David Fabian ,&nbsp;Patrick Morris ,&nbsp;Tibor Szabó","doi":"10.1016/j.jctb.2024.12.006","DOIUrl":"10.1016/j.jctb.2024.12.006","url":null,"abstract":"<div><div>For a graph <em>H</em> and an <em>n</em>-vertex graph <em>G</em>, the <em>H</em>-bootstrap process on <em>G</em> is the process which starts with <em>G</em> and, at every time step, adds any missing edges on the vertices of <em>G</em> that complete a copy of <em>H</em>. This process eventually stabilises and we are interested in the extremal question raised by Bollobás of determining the maximum <em>running time</em> (number of time steps before stabilising) of this process over all possible choices of <em>n</em>-vertex graph <em>G</em>. In this paper, we initiate a systematic study of the asymptotics of this parameter, denoted <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, and its dependence on properties of the graph <em>H</em>. Our focus is on <em>H</em> which define relatively fast bootstrap processes, that is, with <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> being at most linear in <em>n</em>. We study the graph class of trees, showing that one can bound <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> by a quadratic function in <span><math><mi>v</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> for all trees <em>T</em> and all <em>n</em>. We then go on to explore the relationship between the running time of the <em>H</em>-process and the minimum vertex degree and connectivity of <em>H</em>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 44-79"},"PeriodicalIF":1.2,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143127972","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":"Cumulant expansion for counting Eulerian orientations","authors":"Mikhail Isaev ,&nbsp;Brendan D. McKay ,&nbsp;Rui-Ray Zhang","doi":"10.1016/j.jctb.2025.01.002","DOIUrl":"10.1016/j.jctb.2025.01.002","url":null,"abstract":"<div><div>An Eulerian orientation is an orientation of the edges of a graph such that every vertex is balanced: its in-degree equals its out-degree. Counting Eulerian orientations corresponds to the crucial partition function in so-called “ice-type models” in statistical physics and is known to be hard for general graphs. For all graphs with good expansion properties and degrees larger than <span><math><msup><mrow><mi>log</mi></mrow><mrow><mn>8</mn></mrow></msup><mo>⁡</mo><mi>n</mi></math></span>, we derive an asymptotic expansion for this count that approximates it to precision <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>c</mi></mrow></msup><mo>)</mo></math></span> for arbitrarily large <em>c</em>, where <em>n</em> is the number of vertices. The proof relies on a new tail bound for the cumulant expansion of the Laplace transform, which is of independent interest.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 263-314"},"PeriodicalIF":1.2,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}