Journal of Combinatorial Theory Series B最新文献

{"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":"2025-02-21","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":"Tree amalgamations and quasi-isometries","authors":"Matthias Hamann","doi":"10.1016/j.jctb.2025.02.003","DOIUrl":"10.1016/j.jctb.2025.02.003","url":null,"abstract":"<div><div>We investigate the connections between tree amalgamations and quasi-isometries. In particular, we prove that the quasi-isometry type of multi-ended accessible quasi-transitive connected locally finite graphs is determined by the quasi-isometry type of their one-ended factors in any of their terminal factorisations. Our results carry over theorems of Papasoglu and Whyte on quasi-isometries between multi-ended groups to those between multi-ended graphs. In the end, we discuss the impact of our results to a question of Woess.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 83-101"},"PeriodicalIF":1.2,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419816","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":"A counterexample to the coarse Menger conjecture","authors":"Tung Nguyen ,&nbsp;Alex Scott ,&nbsp;Paul Seymour","doi":"10.1016/j.jctb.2025.01.004","DOIUrl":"10.1016/j.jctb.2025.01.004","url":null,"abstract":"<div><div>Menger's well-known theorem from 1927 characterizes when it is possible to find <em>k</em> vertex-disjoint paths between two sets of vertices in a graph <em>G</em>. Recently, Georgakopoulos and Papasoglu and, independently, Albrechtsen, Huynh, Jacobs, Knappe and Wollan conjectured a coarse analogue of Menger's theorem, when the <em>k</em> paths are required to be pairwise at some distance at least <em>d</em>. The result is known for <span><math><mi>k</mi><mo>≤</mo><mn>2</mn></math></span>, but we will show that it is false for all <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, even if <em>G</em> is constrained to have maximum degree at most three. We also give a simpler proof of the result when <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 68-82"},"PeriodicalIF":1.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143403450","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":"Clustered coloring of (path + 2K1)-free graphs on surfaces","authors":"Zdeněk Dvořák","doi":"10.1016/j.jctb.2025.02.001","DOIUrl":"10.1016/j.jctb.2025.02.001","url":null,"abstract":"<div><div>Esperet and Joret proved that planar graphs with bounded maximum degree are 3-colorable with bounded clustering. Liu and Wood asked whether the conclusion holds with the assumption of the bounded maximum degree replaced by assuming that no two vertices have many common neighbors. We answer this question in positive, in the following stronger form: Let <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>″</mo></mrow></msubsup></math></span> be the complete join of two isolated vertices with a path on <em>t</em> vertices. For any surface Σ, a subgraph-closed class of graphs drawn on Σ is 3-choosable with bounded clustering if and only if there exists <em>t</em> such that <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>″</mo></mrow></msubsup></math></span> does not belong to the class.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 45-67"},"PeriodicalIF":1.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394992","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":"Ascending subgraph decomposition","authors":"Kyriakos Katsamaktsis ,&nbsp;Shoham Letzter ,&nbsp;Alexey Pokrovskiy ,&nbsp;Benny Sudakov","doi":"10.1016/j.jctb.2025.01.003","DOIUrl":"10.1016/j.jctb.2025.01.003","url":null,"abstract":"<div><div>A typical theme for many well-known decomposition problems is to show that some obvious necessary conditions for decomposing a graph <em>G</em> into copies of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> are also sufficient. One such problem was posed in 1987, by Alavi, Boals, Chartrand, Erdős, and Oellerman. They conjectured that the edges of every graph with <span><math><mo>(</mo><mtable><mtr><mtd><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></math></span> edges can be decomposed into subgraphs <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> such that each <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> has <em>i</em> edges and is isomorphic to a subgraph of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. In this paper we prove this conjecture for sufficiently large <em>m</em>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"173 ","pages":"Pages 14-44"},"PeriodicalIF":1.2,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386566","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":"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":"2025-02-12","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":"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":"2025-01-24","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}

{"title":"Toward a density Corrádi–Hajnal theorem for degenerate hypergraphs","authors":"Jianfeng Hou ,&nbsp;Caiyun Hu ,&nbsp;Heng Li ,&nbsp;Xizhi Liu ,&nbsp;Caihong Yang ,&nbsp;Yixiao Zhang","doi":"10.1016/j.jctb.2025.01.001","DOIUrl":"10.1016/j.jctb.2025.01.001","url":null,"abstract":"<div><div>Given an <em>r</em>-graph <em>F</em> with <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span>, let <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>F</mi><mo>)</mo></math></span> denote the maximum number of edges in an <em>n</em>-vertex <em>r</em>-graph with at most <em>t</em> pairwise vertex-disjoint copies of <em>F</em>. Extending several old results and complementing prior work <span><span>[34]</span></span> on nondegenerate hypergraphs, we initiate a systematic study on <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>F</mi><mo>)</mo></math></span> for degenerate hypergraphs <em>F</em>.</div><div>For a broad class of degenerate hypergraphs <em>F</em>, we present near-optimal upper bounds for <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>F</mi><mo>)</mo></math></span> when <em>n</em> is sufficiently large and <em>t</em> lies in intervals <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mfrac><mrow><mi>ε</mi><mo>⋅</mo><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>]</mo></math></span>, <span><math><mo>[</mo><mfrac><mrow><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow><mrow><mi>ε</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>,</mo><mi>ε</mi><mi>n</mi><mo>]</mo></math></span>, and <span><math><mo>[</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>ε</mi><mo>)</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>v</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>v</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mfrac><mo>]</mo></math></span>, where <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span> is a constant depending only on <em>F</em>. Our results reveal very different structures for extremal constructions across the three intervals, and we provide characterizations of extremal constructions within the first interval. Additionally, we characterize extremal constructions within the second interval for graphs. Our proof for the first interval also applies to a special class of nondegenerate hypergraphs, including those with undetermined Turán densities, partially improving a result in <span><span>[34]</span></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 221-262"},"PeriodicalIF":1.2,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092870","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":"The next case of Andrásfai's conjecture","authors":"Tomasz Łuczak ,&nbsp;Joanna Polcyn ,&nbsp;Christian Reiher","doi":"10.1016/j.jctb.2024.12.010","DOIUrl":"10.1016/j.jctb.2024.12.010","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> denote the maximum number of edges in a triangle-free graph on <em>n</em> vertices which contains no independent sets larger than <em>s</em>. The behaviour of <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> was first studied by Andrásfai, who conjectured that for <span><math><mi>s</mi><mo>&gt;</mo><mi>n</mi><mo>/</mo><mn>3</mn></math></span> this function is determined by appropriately chosen blow-ups of so called Andrásfai graphs. Moreover, he proved <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>4</mn><mi>n</mi><mi>s</mi><mo>+</mo><mn>5</mn><msup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for <span><math><mi>s</mi><mo>/</mo><mi>n</mi><mo>∈</mo><mo>[</mo><mn>2</mn><mo>/</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>]</mo></math></span> and in earlier work we obtained <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>=</mo><mn>3</mn><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>15</mn><mi>n</mi><mi>s</mi><mo>+</mo><mn>20</mn><msup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for <span><math><mi>s</mi><mo>/</mo><mi>n</mi><mo>∈</mo><mo>[</mo><mn>3</mn><mo>/</mo><mn>8</mn><mo>,</mo><mn>2</mn><mo>/</mo><mn>5</mn><mo>]</mo></math></span>. Here we make the next step in the quest to settle Andrásfai's conjecture by proving <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>=</mo><mn>6</mn><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>32</mn><mi>n</mi><mi>s</mi><mo>+</mo><mn>44</mn><msup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for <span><math><mi>s</mi><mo>/</mo><mi>n</mi><mo>∈</mo><mo>[</mo><mn>4</mn><mo>/</mo><mn>11</mn><mo>,</mo><mn>3</mn><mo>/</mo><mn>8</mn><mo>]</mo></math></span>.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"172 ","pages":"Pages 198-220"},"PeriodicalIF":1.2,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092869","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":"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":"2025-01-13","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}