{"title":"Giant rainbow trees in sparse random graphs","authors":"Tolson Bell , Alan Frieze","doi":"10.1016/j.ejc.2025.104184","DOIUrl":"10.1016/j.ejc.2025.104184","url":null,"abstract":"<div><div>For any small constant <span><math><mrow><mi>ϵ</mi><mo>></mo><mn>0</mn></mrow></math></span>, the Erdős-Rényi random graph <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>ϵ</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo>)</mo></mrow></mrow></math></span> with high probability has a unique largest component which contains <span><math><mrow><mrow><mo>(</mo><mn>1</mn><mo>±</mo><mi>O</mi><mrow><mo>(</mo><mi>ϵ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mn>2</mn><mi>ϵ</mi><mi>n</mi></mrow></math></span> vertices. Let <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> be obtained by assigning each edge in <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></span> a color in <span><math><mrow><mo>[</mo><mi>c</mi><mo>]</mo></mrow></math></span> independently and uniformly. Cooley, Do, Erde, and Missethan proved that for any fixed <span><math><mrow><mi>α</mi><mo>></mo><mn>0</mn></mrow></math></span>, <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>α</mi><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>ϵ</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo>)</mo></mrow></mrow></math></span> with high probability contains a rainbow tree (a tree that does not repeat colors) which covers <span><math><mrow><mrow><mo>(</mo><mn>1</mn><mo>±</mo><mi>O</mi><mrow><mo>(</mo><mi>ϵ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mfrac><mrow><mi>α</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mi>ϵ</mi><mi>n</mi></mrow></math></span> vertices, and conjectured that there is one which covers <span><math><mrow><mrow><mo>(</mo><mn>1</mn><mo>±</mo><mi>O</mi><mrow><mo>(</mo><mi>ϵ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mn>2</mn><mi>ϵ</mi><mi>n</mi></mrow></math></span>. In this paper, we achieve the correct leading constant and prove their conjecture correct up to a logarithmic factor in the error term, as we show that with high probability <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>α</mi><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>ϵ</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo>)</mo></mrow></mrow></math></span> contains a rainbow tree which covers <span><math><mrow><mrow><mo>(</mo><mn>1</mn><mo>±</mo><mi>O</mi><mrow><mo>(</mo><mi>ϵ</mi><mo>log</mo><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mi>ϵ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mn>2</mn><mi>ϵ</mi><mi>n</mi></mrow></math></span> vertices.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"128 ","pages":"Article 104184"},"PeriodicalIF":1.0,"publicationDate":"2025-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144131759","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}
{"title":"A generalization of the Erdős-Birch theorem","authors":"Wang-Xing Yu , Yong-Gao Chen , Shi-Qiang Chen","doi":"10.1016/j.ejc.2025.104187","DOIUrl":"10.1016/j.ejc.2025.104187","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> denote the set of all nonnegative integers. A set <span><math><mi>T</mi></math></span> of positive integers is called complete if every sufficiently large integer is the sum of distinct integers taken from <span><math><mi>T</mi></math></span>. In 1959, Birch confirmed a conjecture of Erdős by proving that the set <span><math><mrow><mo>{</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><msubsup><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>:</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>}</mo></mrow></math></span> is complete if <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>></mo><mn>1</mn></mrow></math></span> are integers with <span><math><mrow><mo>gcd</mo><mrow><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span>. In this paper, the following result is proved: if <span><math><mrow><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>k</mi></mrow></msub><mo>></mo><mn>1</mn></mrow></math></span> are integers, then the set <span><span><span><math><mrow><mo>{</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>⋯</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msubsup><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>k</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>}</mo></mrow></math></span></span></span>is complete if and only if <span><math><mrow><mo>gcd</mo><mrow><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>k</mi></mrow></msub><mo>)</mo></mrow><mo>≤</mo><mn>2</mn></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"128 ","pages":"Article 104187"},"PeriodicalIF":1.0,"publicationDate":"2025-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144131758","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}
Zdeněk Dvořák, Benjamin Moore , Michaela Seifrtová, Robert Šámal
{"title":"Precoloring extension in planar near-Eulerian-triangulations","authors":"Zdeněk Dvořák, Benjamin Moore , Michaela Seifrtová, Robert Šámal","doi":"10.1016/j.ejc.2025.104183","DOIUrl":"10.1016/j.ejc.2025.104183","url":null,"abstract":"<div><div>We consider the 4-precoloring extension problem in <em>planar near-Eulerian- triangulations</em>, i.e., plane graphs where all faces except possibly for the outer one have length three, all vertices not incident with the outer face have even degree, and exactly the vertices incident with the outer face are precolored. We give a necessary topological condition for the precoloring to extend, and give a complete characterization when the outer face has length at most five and when all vertices of the outer face have odd degree and are colored using only three colors.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"129 ","pages":"Article 104183"},"PeriodicalIF":1.0,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144588464","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}
{"title":"Combinatorial interpretations of truncated series from the Jacobi triple product identity","authors":"Olivia X.M. Yao","doi":"10.1016/j.ejc.2025.104176","DOIUrl":"10.1016/j.ejc.2025.104176","url":null,"abstract":"<div><div>In their seminal work on truncated sums of theta series, Andrews and Merca, and Guo and Zeng independently posed a conjecture on the sign of the coefficients of truncated sums of Jacobi’s triple product identity. This conjecture was confirmed independently by Mao by utilizing analytic method and by Yee by using combinatorial method. In 2019, Wang and Yee reproved this conjecture by establishing an explicit series with nonnegative coefficients and proved a companion identity. In this paper, we present partition-theoretic interpretations of the truncated sums posed by Andrews and Merca and Guo and Zeng, and Wang and Yee. We determine what partitions of <span><math><mi>n</mi></math></span> are counted by the truncated sums based on the minimal excludant in congruence classes of partitions. As applications, we settle two open problems on partition-theoretic interpretations of series posed by Guo and Zeng, and Merca, respectively.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"128 ","pages":"Article 104176"},"PeriodicalIF":1.0,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143946618","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}
Alan J. Cain , António Malheiro , Fátima Rodrigues , Inês Rodrigues
{"title":"A local characterization of quasi-crystal graphs","authors":"Alan J. Cain , António Malheiro , Fátima Rodrigues , Inês Rodrigues","doi":"10.1016/j.ejc.2025.104172","DOIUrl":"10.1016/j.ejc.2025.104172","url":null,"abstract":"<div><div>A local characterization of quasi-crystal graphs of type <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> is provided, by presenting a set of local axioms, similar to the ones introduced by Stembridge for crystal graphs of simply-laced root systems, but restricted to type <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>. It is also shown that quasi-crystal graphs satisfying these axioms are closed under the tensor product recently introduced by Cain, Guilherme and Malheiro. It is deduced that each connected component of such a graph has a unique highest weight element, whose weight is a composition, and it is isomorphic to a quasi-crystal graph of semistandard quasi-ribbon tableaux.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"128 ","pages":"Article 104172"},"PeriodicalIF":1.0,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144070090","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}
{"title":"Sparsity of 3-flow-critical graphs","authors":"Zdeněk Dvořák , Sergey Norin","doi":"10.1016/j.ejc.2025.104174","DOIUrl":"10.1016/j.ejc.2025.104174","url":null,"abstract":"<div><div>A connected graph <span><math><mi>G</mi></math></span> is 3-flow-critical if <span><math><mi>G</mi></math></span> does not have a nowhere-zero 3-flow, but every proper contraction of <span><math><mi>G</mi></math></span> does. We prove that every <span><math><mi>n</mi></math></span>-vertex 3-flow-critical graph other than <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> has at least <span><math><mrow><mfrac><mrow><mn>5</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mi>n</mi></mrow></math></span> edges. This bound is tight up to lower-order terms, answering a question of Li et al. (2022). It also generalizes the result of Koester (1991) on the maximum average degree of 4-critical planar graphs.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"128 ","pages":"Article 104174"},"PeriodicalIF":1.0,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143942228","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}
{"title":"On the cop number and the weak Meyniel conjecture for algebraic graphs","authors":"Arindam Biswas , Jyoti Prakash Saha","doi":"10.1016/j.ejc.2025.104168","DOIUrl":"10.1016/j.ejc.2025.104168","url":null,"abstract":"<div><div>We show that the cop number of the Cayley sum graph of a finite group <span><math><mi>G</mi></math></span> with respect to a symmetric subset <span><math><mi>S</mi></math></span> is at most twice its degree when the graph is connected, undirected. We also prove that a similar bound holds for the cop number of generalised Cayley graphs and twisted Cayley sum graphs under some conditions. These extend a result of Frankl to such graphs. Using the above bounds and a result of Bollobás–Janson–Riordan, we show that the weak Meyniel conjecture holds for these algebraic graphs.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"128 ","pages":"Article 104168"},"PeriodicalIF":1.0,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143934942","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}
{"title":"Infinitely many minimally non-Ramsey size-linear graphs","authors":"Yuval Wigderson","doi":"10.1016/j.ejc.2025.104175","DOIUrl":"10.1016/j.ejc.2025.104175","url":null,"abstract":"<div><div>A graph <span><math><mi>G</mi></math></span> is said to be Ramsey size-linear if <span><math><mrow><mi>r</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>e</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> for every graph <span><math><mi>H</mi></math></span> with no isolated vertices. Erdős, Faudree, Rousseau, and Schelp observed that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> is not Ramsey size-linear, but each of its proper subgraphs is, and they asked whether there exist infinitely many such graphs. In this short note, we answer this question in the affirmative.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"128 ","pages":"Article 104175"},"PeriodicalIF":1.0,"publicationDate":"2025-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143928353","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}
{"title":"Subspaces, subsets, and Motzkin paths","authors":"Jonathan D. Farley , Murali K. Srinivasan","doi":"10.1016/j.ejc.2025.104173","DOIUrl":"10.1016/j.ejc.2025.104173","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> denote the set of all Motzkin paths from <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></math></span> to <span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></math></span>. For each <span><math><mrow><mi>P</mi><mo>∈</mo><mi>M</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> we define a statistic <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>P</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span>, the weight of <span><math><mi>P</mi></math></span>. Let <span><math><mrow><mo>|</mo><mi>P</mi><mo>|</mo></mrow></math></span> denote the number of down steps in <span><math><mrow><mi>P</mi><mo>∈</mo><mi>M</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>. Let <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> denote the projective geometry (= poset of subspaces of an <span><math><mi>n</mi></math></span>-dimensional vector space over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>). We define a map from <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> to <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> and show that, for <span><math><mrow><mi>P</mi><mo>∈</mo><mi>M</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, the inverse image of <span><math><mi>P</mi></math></span> consists of a disjoint union of <span><math><mrow><msup><mrow><mrow><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mrow><mo>|</mo><mi>P</mi><mo>|</mo></mrow></mrow></msup><mi>w</mi><mrow><mo>(</mo><mi>P</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> symmetric Boolean subsets in <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, all with minimum rank <span><math><mrow><mo>|</mo><mi>P</mi><mo>|</mo></mrow></math></span> and maximum rank <span><math><mrow><mi>n</mi><mo>−</mo><mrow><mo>|</mo><mi>P</mi><mo>|</mo></mrow></mrow></math></span>. This yields an explicit symmetric Boolean decomposition of the projective geometry and gives a poset theoretic interpretation to the identity <span><span><span><math><mrow><msub><mrow><mfenced><mfrac><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></mfrac></mfenced></mrow><mrow><mi>q</mi></mrow></msub><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><mi>P</mi><mo>∈</mo><mi>M</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></munder><msup><mrow><mrow><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mrow><mo>|</mo><mi>P</mi><mo>|</mo></mrow></mrow></msup><mi>w</mi><mrow><mo>(</mo><mi>P</mi><mo>,</mo><mi>q</mi><mo>)</mo></","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"128 ","pages":"Article 104173"},"PeriodicalIF":1.0,"publicationDate":"2025-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143928354","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}
{"title":"Degree-balanced decompositions of cubic graphs","authors":"Borut Lužar , Jakub Przybyło , Roman Soták","doi":"10.1016/j.ejc.2025.104169","DOIUrl":"10.1016/j.ejc.2025.104169","url":null,"abstract":"<div><div>We show that every cubic graph on <span><math><mi>n</mi></math></span> vertices contains a spanning subgraph in which the number of vertices of each degree deviates from <span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></mfrac></math></span> by at most <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, up to three exceptions. This resolves the conjecture of Alon and Wei (2023) for cubic graphs.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"128 ","pages":"Article 104169"},"PeriodicalIF":1.0,"publicationDate":"2025-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143928518","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}
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