European Journal of Combinatorics最新文献

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Distributions of parity differences and biases in partitions into distinct parts
IF 1 3区 数学
European Journal of Combinatorics Pub Date : 2025-04-03 DOI: 10.1016/j.ejc.2025.104157
Siu Hang Man
{"title":"Distributions of parity differences and biases in partitions into distinct parts","authors":"Siu Hang Man","doi":"10.1016/j.ejc.2025.104157","DOIUrl":"10.1016/j.ejc.2025.104157","url":null,"abstract":"<div><div>For a partition <span><math><mrow><mi>λ</mi><mo>⊢</mo><mi>n</mi></mrow></math></span>, we let <span><math><mrow><mo>pd</mo><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></math></span>, the parity difference of <span><math><mi>λ</mi></math></span>, be the number of odd parts of <span><math><mi>λ</mi></math></span> minus the number of even parts of <span><math><mi>λ</mi></math></span>. We prove for <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>R</mi></mrow></math></span> an asymptotic expansion for the number of partitions of <span><math><mi>n</mi></math></span> into distinct parts with normalised parity difference <span><math><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo>pd</mo><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></math></span> greater than <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> as <span><math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math></span>. As a corollary, we find the distribution of the parity differences and parity biases for partitions of <span><math><mi>n</mi></math></span> into distinct parts. We also establish analogous results for generalised parity differences modulo <span><math><mi>N</mi></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104157"},"PeriodicalIF":1.0,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Extremal problems for intersecting families of subspaces with a measure
IF 1 3区 数学
European Journal of Combinatorics Pub Date : 2025-03-28 DOI: 10.1016/j.ejc.2025.104156
Hajime Tanaka , Norihide Tokushige
{"title":"Extremal problems for intersecting families of subspaces with a measure","authors":"Hajime Tanaka ,&nbsp;Norihide Tokushige","doi":"10.1016/j.ejc.2025.104156","DOIUrl":"10.1016/j.ejc.2025.104156","url":null,"abstract":"<div><div>We introduce a measure for subspaces of a vector space over a <span><math><mi>q</mi></math></span>-element field, and propose some extremal problems for intersecting families. These are <span><math><mi>q</mi></math></span>-analogue of Erdős–Ko–Rado type problems, and we answer some of the basic questions.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104156"},"PeriodicalIF":1.0,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143714877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On hypergraph Turán problems with bounded matching number
IF 1 3区 数学
European Journal of Combinatorics Pub Date : 2025-03-26 DOI: 10.1016/j.ejc.2025.104155
Dániel Gerbner , Casey Tompkins , Junpeng Zhou
{"title":"On hypergraph Turán problems with bounded matching number","authors":"Dániel Gerbner ,&nbsp;Casey Tompkins ,&nbsp;Junpeng Zhou","doi":"10.1016/j.ejc.2025.104155","DOIUrl":"10.1016/j.ejc.2025.104155","url":null,"abstract":"<div><div>Very recently, Alon and Frankl, and Gerbner studied the maximum number of edges in <span><math><mi>n</mi></math></span>-vertex <span><math><mi>F</mi></math></span>-free graphs with bounded matching number, respectively. We consider the analogous Turán problems on hypergraphs with bounded matching number, and we obtain some exact results.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104155"},"PeriodicalIF":1.0,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The emergence of a giant rainbow component
IF 1 3区 数学
European Journal of Combinatorics Pub Date : 2025-03-25 DOI: 10.1016/j.ejc.2025.104154
Oliver Cooley , Tuan Anh Do , Joshua Erde , Michael Missethan
{"title":"The emergence of a giant rainbow component","authors":"Oliver Cooley ,&nbsp;Tuan Anh Do ,&nbsp;Joshua Erde ,&nbsp;Michael Missethan","doi":"10.1016/j.ejc.2025.104154","DOIUrl":"10.1016/j.ejc.2025.104154","url":null,"abstract":"<div><div>The <em>random coloured graph</em> <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></span> is obtained from the Erdős–Rényi binomial random graph <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></span> by assigning to each edge a colour from a set of <span><math><mi>c</mi></math></span> colours independently and uniformly at random. It is not hard to see that, when <span><math><mrow><mi>c</mi><mo>=</mo><mi>Θ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, the order of the largest rainbow tree in this model undergoes a phase transition at the critical point <span><math><mrow><mi>p</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></math></span>. In this paper, we determine the asymptotic order of the largest rainbow tree in the <em>weakly sub- and supercritical regimes</em>, when <span><math><mrow><mi>p</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>ɛ</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></math></span> for some <span><math><mrow><mi>ɛ</mi><mo>=</mo><mi>ɛ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> which satisfies <span><math><mrow><mi>ɛ</mi><mo>=</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mrow><mrow><mo>|</mo><mi>ɛ</mi><mo>|</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math></span>. In particular, we show that in both of these regimes with high probability the largest component of <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></span> contains a rainbow tree which is almost spanning. We also consider the order of the largest rainbow tree in the <em>sparse regime</em>, when <span><math><mrow><mi>p</mi><mo>=</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></math></span> for some constant <span><math><mrow><mi>d</mi><mo>&gt;</mo><mn>1</mn></mrow></math></span>. Here we show that the largest rainbow tree has linear order, and, moreover, for <span><math><mi>d</mi></math></span> and <span><math><mi>c</mi></math></span> sufficiently large, with high probability <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></span> even contains a rainbow cycle which is almost spanning.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104154"},"PeriodicalIF":1.0,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Extensions of transversal valuated matroids
IF 1 3区 数学
European Journal of Combinatorics Pub Date : 2025-03-19 DOI: 10.1016/j.ejc.2025.104153
Alex Fink , Jorge Alberto Olarte
{"title":"Extensions of transversal valuated matroids","authors":"Alex Fink ,&nbsp;Jorge Alberto Olarte","doi":"10.1016/j.ejc.2025.104153","DOIUrl":"10.1016/j.ejc.2025.104153","url":null,"abstract":"<div><div>Following up on our previous work, we study single-element extensions of transversal valuated matroids. We show that tropical presentations of valuated matroids with a minimal set of finite entries enjoy counterparts of the properties proved by Bonin and de Mier of minimal non-valuated transversal presentations.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104153"},"PeriodicalIF":1.0,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Minors, connectivity, and diameter in randomly perturbed sparse graphs
IF 1 3区 数学
European Journal of Combinatorics Pub Date : 2025-03-17 DOI: 10.1016/j.ejc.2025.104152
Elad Aigner-Horev , Dan Hefetz , Michael Krivelevich
{"title":"Minors, connectivity, and diameter in randomly perturbed sparse graphs","authors":"Elad Aigner-Horev ,&nbsp;Dan Hefetz ,&nbsp;Michael Krivelevich","doi":"10.1016/j.ejc.2025.104152","DOIUrl":"10.1016/j.ejc.2025.104152","url":null,"abstract":"<div><div>Extremal properties of sparse graphs, randomly perturbed by the binomial random graph are considered. It is known that every <span><math><mi>n</mi></math></span>-vertex graph <span><math><mi>G</mi></math></span> contains a complete minor of order <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><mi>n</mi><mo>/</mo><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>. We prove that adding <span><math><mrow><mi>ξ</mi><mi>n</mi></mrow></math></span> random edges, where <span><math><mrow><mi>ξ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> is arbitrarily small yet fixed, to an <span><math><mi>n</mi></math></span>-vertex graph <span><math><mi>G</mi></math></span> satisfying <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>ζ</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mi>n</mi></mrow></math></span> asymptotically almost surely results in a graph containing a complete minor of order <span><math><mrow><mover><mrow><mi>Ω</mi></mrow><mrow><mo>̃</mo></mrow></mover><mfenced><mrow><mi>n</mi><mo>/</mo><msqrt><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msqrt></mrow></mfenced></mrow></math></span>; this result is tight up to the implicit logarithmic terms.</div><div>For complete topological minors, we prove that there exists a constant <span><math><mrow><mi>C</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> such that adding <span><math><mrow><mi>C</mi><mi>n</mi></mrow></math></span> random edges to a graph <span><math><mi>G</mi></math></span> satisfying <span><math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>ω</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, asymptotically almost surely results in a graph containing a complete topological minor of order <span><math><mrow><mover><mrow><mi>Ω</mi></mrow><mrow><mo>̃</mo></mrow></mover><mrow><mo>(</mo><mo>min</mo><mrow><mo>{</mo><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>; this result is tight up to the implicit logarithmic terms.</div><div>Finally, extending results of Bohman, Frieze, Krivelevich, and Martin for the dense case, we analyse the asymptotic behaviour of the vertex-connectivity and the diameter of randomly perturbed sparse graphs.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104152"},"PeriodicalIF":1.0,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143631754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Generating functions for fixed points of the Mullineux map 穆利纽克斯图定点的生成函数
IF 1 3区 数学
European Journal of Combinatorics Pub Date : 2025-03-14 DOI: 10.1016/j.ejc.2025.104141
David J. Hemmer
{"title":"Generating functions for fixed points of the Mullineux map","authors":"David J. Hemmer","doi":"10.1016/j.ejc.2025.104141","DOIUrl":"10.1016/j.ejc.2025.104141","url":null,"abstract":"<div><div>Mullineux defined an involution on the set of <span><math><mi>e</mi></math></span>-regular partitions of <span><math><mi>n</mi></math></span>. When <span><math><mrow><mi>e</mi><mo>=</mo><mi>p</mi></mrow></math></span> is prime, these partitions label irreducible symmetric group modules in characteristic <span><math><mi>p</mi></math></span>. Mullineux’s conjecture, since proven, was that this “Mullineux map” described the effect on the labels of taking the tensor product with the one-dimensional signature representation. Counting irreducible modules fixed by this tensor product is related to counting irreducible modules for the alternating group <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> in prime characteristic. In 1991, Andrews and Olsson worked out the generating function counting fixed points of Mullineux’s map when <span><math><mrow><mi>e</mi><mo>=</mo><mi>p</mi></mrow></math></span> is an odd prime (providing evidence in support of Mullineux’s conjecture). In 1998, Bessenrodt and Olsson counted the fixed points in a <span><math><mi>p</mi></math></span>-block of weight <span><math><mi>w</mi></math></span>. We extend both results to arbitrary <span><math><mi>e</mi></math></span>, and determine the corresponding generating functions. When <span><math><mi>e</mi></math></span> is odd but not prime the extension is immediate, while <span><math><mi>e</mi></math></span> even requires additional work and the results, which are different, have not appeared in the literature.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104141"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143629043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A refinement on spectral Mantel’s theorem
IF 1 3区 数学
European Journal of Combinatorics Pub Date : 2025-03-14 DOI: 10.1016/j.ejc.2025.104142
Zhenzhen Lou , Lu Lu , Mingqing Zhai
{"title":"A refinement on spectral Mantel’s theorem","authors":"Zhenzhen Lou ,&nbsp;Lu Lu ,&nbsp;Mingqing Zhai","doi":"10.1016/j.ejc.2025.104142","DOIUrl":"10.1016/j.ejc.2025.104142","url":null,"abstract":"<div><div>The well-known Mantel’s theorem states that <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>⌊</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>4</mn><mo>⌋</mo></mrow></mrow></math></span> for every <span><math><mi>n</mi></math></span>-vertex triangle-free graph <span><math><mi>G</mi></math></span>. In 1970, Nosal showed a spectral version of Mantel’s theorem, which states that <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><msqrt><mrow><mi>m</mi></mrow></msqrt></mrow></math></span> for every triangle-free graph <span><math><mi>G</mi></math></span> on <span><math><mi>m</mi></math></span> edges. Later, Nikiforov proved that the equality holds in Nosal’s bound if and only if <span><math><mi>G</mi></math></span> is a complete bipartite graph. Lin, Ning, and Wu [Combin. Probab. Comput. 30 (2021)] first gave a result on non-bipartite triangle-free graphs. They proved that <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><msqrt><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msqrt></mrow></math></span>, and equality holds if and only if <span><math><mi>G</mi></math></span> is a 5-cycle. Their result is actually stronger than Mantel’s theorem. Recently, Li, Feng and Peng [Electron. J. Combin. 31 (2024)] characterized the extremal non-bipartite triangle-free graphs with maximal spectral radius for even <span><math><mi>m</mi></math></span>. Furthermore, they proposed a question as follows: what is the extremal graph with maximal spectral radius over all non-bipartite <span><math><mrow><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo></mrow></math></span>-free graphs of even size <span><math><mi>m</mi></math></span>? This question is completely solved in this paper. Here we use a different technique, and we call it residual function, which can present more concise proofs on related problems. Finally, a general question is proposed for further research.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104142"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143629042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Tight bound on the minimum degree to guarantee graphs forbidding some odd cycles to be bipartite 保证禁止某些奇数循环的图为两部分图的最小度数的严格约束
IF 1 3区 数学
European Journal of Combinatorics Pub Date : 2025-03-14 DOI: 10.1016/j.ejc.2025.104143
Xiaoli Yuan, Yuejian Peng
{"title":"Tight bound on the minimum degree to guarantee graphs forbidding some odd cycles to be bipartite","authors":"Xiaoli Yuan,&nbsp;Yuejian Peng","doi":"10.1016/j.ejc.2025.104143","DOIUrl":"10.1016/j.ejc.2025.104143","url":null,"abstract":"<div><div>Erdős and Simonovits asked the following question: For an integer <span><math><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and a family of non-bipartite graphs <span><math><mi>H</mi></math></span>, determine the infimum of <span><math><mi>α</mi></math></span> such that any <span><math><mi>H</mi></math></span>-free <span><math><mi>n</mi></math></span>-vertex graph with minimum degree at least <span><math><mrow><mi>α</mi><mi>n</mi></mrow></math></span> has chromatic number at most <span><math><mi>r</mi></math></span>. We answer this question for <span><math><mrow><mi>r</mi><mo>=</mo><mn>2</mn></mrow></math></span> and any family consisting of odd cycles. Let <span><math><mi>C</mi></math></span> be a family of odd cycles in which <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is the shortest odd cycle not in <span><math><mi>C</mi></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is the longest odd cycle in <span><math><mi>C</mi></math></span>, we show that if <span><math><mi>G</mi></math></span> is an <span><math><mi>n</mi></math></span>-vertex <span><math><mi>C</mi></math></span>-free graph with <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1000</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>8</mn></mrow></msup></mrow></math></span> and <span><math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>&gt;</mo><mo>max</mo><mrow><mo>{</mo><mi>n</mi><mo>/</mo><mrow><mo>(</mo><mn>2</mn><mrow><mo>(</mo><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo><mn>2</mn><mi>n</mi><mo>/</mo><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>, then <span><math><mi>G</mi></math></span> is bipartite. Moreover, this bound on the minimum degree is tight.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104143"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143628533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Precoloring extension in planar near-Eulerian-triangulations
IF 1 3区 数学
European Journal of Combinatorics Pub Date : 2025-03-07 DOI: 10.1016/j.ejc.2025.104138
Zdeněk Dvořák, Benjamin Moore , Michaela Seifrtová, Robert Šámal
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