Combinatorica最新文献_第5页

Arc Connectivity and Submodular Flows in Digraphs 数图中的弧连接和次模流

IF 1.1 2区数学

Combinatorica Pub Date : 2024-05-28 DOI: 10.1007/s00493-024-00108-0

Ahmad Abdi, Gérard Cornuéjols, Giacomo Zambelli

{"title":"Arc Connectivity and Submodular Flows in Digraphs","authors":"Ahmad Abdi, Gérard Cornuéjols, Giacomo Zambelli","doi":"10.1007/s00493-024-00108-0","DOIUrl":"https://doi.org/10.1007/s00493-024-00108-0","url":null,"abstract":"Let (D=(V,A)) be a digraph. For an integer (kge 1), a k-arc-connected flip is an arc subset of D such that after reversing the arcs in it the digraph becomes (strongly) k-arc-connected. The first main result of this paper introduces a sufficient condition for the existence of a k-arc-connected flip that is also a submodular flow for a crossing submodular function. More specifically, given some integer (tau ge 1), suppose (d_A^+(U)+(frac{tau }{k}-1)d_A^-(U)ge tau ) for all (Usubsetneq V, Une emptyset ), where (d_A^+(U)) and (d_A^-(U)) denote the number of arcs in A leaving and entering U, respectively. Let ({mathcal {C}}) be a crossing family over ground set V, and let (f:{mathcal {C}}rightarrow {mathbb {Z}}) be a crossing submodular function such that (f(U)ge frac{k}{tau }(d_A^+(U)-d_A^-(U))) for all (Uin {mathcal {C}}). Then D has a k-arc-connected flip J such that (f(U)ge d_J^+(U)-d_J^-(U)) for all (Uin {mathcal {C}}). The result has several applications to Graph Orientations and Combinatorial Optimization. In particular, it strengthens Nash-Williams’ so-called weak orientation theorem, and proves a weaker variant of Woodall’s conjecture on digraphs whose underlying undirected graph is (tau )-edge-connected. The second main result of this paper is even more general. It introduces a sufficient condition for the existence of capacitated integral solutions to the intersection of two submodular flow systems. This sufficient condition implies the classic result of Edmonds and Giles on the box-total dual integrality of a submodular flow system. It also has the consequence that in a weakly connected digraph, the intersection of two submodular flow systems is totally dual integral.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"29 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141159422","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

Effective Bounds for Induced Size-Ramsey Numbers of Cycles 诱导大小拉姆齐循环数的有效界限

IF 1.1 2区数学

Combinatorica Pub Date : 2024-05-14 DOI: 10.1007/s00493-024-00103-5

Domagoj Bradač, Nemanja Draganić, Benny Sudakov

{"title":"Effective Bounds for Induced Size-Ramsey Numbers of Cycles","authors":"Domagoj Bradač, Nemanja Draganić, Benny Sudakov","doi":"10.1007/s00493-024-00103-5","DOIUrl":"https://doi.org/10.1007/s00493-024-00103-5","url":null,"abstract":"The induced size-Ramsey number (hat{r}_text {ind}^k(H)) of a graph H is the smallest number of edges a (host) graph G can have such that for any k-coloring of its edges, there exists a monochromatic copy of H which is an induced subgraph of G. In 1995, in their seminal paper, Haxell, Kohayakawa and Łuczak showed that for cycles, these numbers are linear for any constant number of colours, i.e., (hat{r}_text {ind}^k(C_n)le Cn) for some (C=C(k)). The constant C comes from the use of the regularity lemma, and has a tower type dependence on k. In this paper we significantly improve these bounds, showing that (hat{r}_text {ind}^k(C_n)le O(k^{102})n) when n is even, thus obtaining only a polynomial dependence of C on k. We also prove (hat{r}_text {ind}^k(C_n)le e^{O(klog k)}n) for odd n, which almost matches the lower bound of (e^{Omega (k)}n). Finally, we show that the ordinary (non-induced) size-Ramsey number satisfies (hat{r}^k(C_n)=e^{O(k)}n) for odd n. This substantially improves the best previous result of (e^{O(k^2)}n), and is best possible, up to the implied constant in the exponent. To achieve our results, we present a new host graph construction which, roughly speaking, reduces our task to finding a cycle of approximate given length in a graph with local sparsity.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"67 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140919474","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

A Proof of a Frankl–Kupavskii Conjecture on Intersecting Families 弗兰克尔-库帕夫斯基相交族猜想的证明

IF 1.1 2区数学

Combinatorica Pub Date : 2024-05-14 DOI: 10.1007/s00493-024-00105-3

Agnijo Banerjee

引用次数: 0

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