Combinatorica最新文献_第4页

Criticality in Sperner’s Lemma 斯佩尔纳定理中的临界点

IF 1.1 2区数学

Combinatorica Pub Date : 2024-05-14 DOI: 10.1007/s00493-024-00104-4

Tomáš Kaiser, Matěj Stehlík, Riste Škrekovski

{"title":"Criticality in Sperner’s Lemma","authors":"Tomáš Kaiser, Matěj Stehlík, Riste Škrekovski","doi":"10.1007/s00493-024-00104-4","DOIUrl":"https://doi.org/10.1007/s00493-024-00104-4","url":null,"abstract":"We answer a question posed by Gallai in 1969 concerning criticality in Sperner’s lemma, listed as Problem 9.14 in the collection of Jensen and Toft (Graph coloring problems, Wiley, New York, 1995). Sperner’s lemma states that if a labelling of the vertices of a triangulation of the d-simplex (Delta ^d) with labels (1, 2, ldots , d+1) has the property that (i) each vertex of (Delta ^d) receives a distinct label, and (ii) any vertex lying in a face of (Delta ^d) has the same label as one of the vertices of that face, then there exists a rainbow facet (a facet whose vertices have pairwise distinct labels). For (dle 2), it is not difficult to show that for every facet (sigma ), there exists a labelling with the above properties where (sigma ) is the unique rainbow facet. For every (dge 3), however, we construct an infinite family of examples where this is not the case, which implies the answer to Gallai’s question as a corollary. The construction is based on the properties of a 4-polytope which had been used earlier to disprove a claim of Motzkin on neighbourly polytopes.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"45 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 the Zarankiewicz Problem for Graphs with Bounded VC-Dimension 关于有界 VC 维度图的扎兰凯维奇问题

IF 1.1 2区数学

Combinatorica Pub Date : 2024-05-14 DOI: 10.1007/s00493-024-00095-2

Oliver Janzer, Cosmin Pohoata

{"title":"On the Zarankiewicz Problem for Graphs with Bounded VC-Dimension","authors":"Oliver Janzer, Cosmin Pohoata","doi":"10.1007/s00493-024-00095-2","DOIUrl":"https://doi.org/10.1007/s00493-024-00095-2","url":null,"abstract":"The problem of Zarankiewicz asks for the maximum number of edges in a bipartite graph on n vertices which does not contain the complete bipartite graph (K_{k,k}) as a subgraph. A classical theorem due to Kővári, Sós, and Turán says that this number of edges is (Oleft( n^{2 - 1/k}right) ). An important variant of this problem is the analogous question in bipartite graphs with VC-dimension at most d, where d is a fixed integer such that (k ge d ge 2). A remarkable result of Fox et al. (J. Eur. Math. Soc. (JEMS) 19:1785–1810, 2017) with multiple applications in incidence geometry shows that, under this additional hypothesis, the number of edges in a bipartite graph on n vertices and with no copy of (K_{k,k}) as a subgraph must be (Oleft( n^{2 - 1/d}right) ). This theorem is sharp when (k=d=2), because by design any (K_{2,2})-free graph automatically has VC-dimension at most 2, and there are well-known examples of such graphs with (Omega left( n^{3/2}right) ) edges. However, it turns out this phenomenon no longer carries through for any larger d. We show the following improved result: the maximum number of edges in bipartite graphs with no copies of (K_{k,k}) and VC-dimension at most d is (o(n^{2-1/d})), for every (k ge d ge 3).\u0000","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"34 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140919414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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

List-k-Coloring H-Free Graphs for All $$k>4$$ 针对所有 $$k>4$$ 的列表-k-着色无 H 图

IF 1.1 2区数学

Combinatorica Pub Date : 2024-05-14 DOI: 10.1007/s00493-024-00106-2

Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl

引用次数: 0

Rainbow Cycles in Properly Edge-Colored Graphs 适当边缘着色图形中的彩虹循环

IF 1.1 2区数学

Combinatorica Pub Date : 2024-05-02 DOI: 10.1007/s00493-024-00101-7

Jaehoon Kim, Joonkyung Lee, Hong Liu, Tuan Tran

引用次数: 0

Rainbow Variations on a Theme by Mantel: Extremal Problems for Gallai Colouring Templates 曼特尔的主题彩虹变奏曲：加莱填色模板的极值问题