Advances in Combinatorics最新文献

{"title":"Extremal functions for sparse minors","authors":"Kevin Hendrey, S. Norin, D. Wood","doi":"10.19086/aic.2022.5","DOIUrl":"https://doi.org/10.19086/aic.2022.5","url":null,"abstract":"The notion of a graph minor, which generalizes graph subgraphs, is a central notion of modern graph theory. Classical results concerning graph minors include the Graph Minor Theorem and the Graph Structure Theorem, both due to Robertson and Seymour. The results concern properties of classes of graphs closed under taking minors; such graph classes include many important natural classes of graphs, e.g., the class of planar graphs and, more generally, the class of graphs embeddable in a fixed surface.\u0000\u0000The Graph Minor Theorem asserts that every class of graphs closed under taking minors has a finite list of forbidden minors. For example, Wagner’s Theorem, which claims that a graph is planar if and only if it does not contain or as a minor, is a particular case of this theorem. The Graph Structure Theorem asserts that graphs from a fixed class of graphs closed under taking minors can be decomposed in a tree-like fashion into graphs almost embeddable in a fixed surface. In particular, every graph in a class of graphs avoiding a fixed minor admits strongly sublinear separators (the Planar separator theorem of Lipton and Tarjan is a special case of this more general result).\u0000\u0000As the number of edges of every graph contained in a class of graphs closed under taking minors is linear in the number of its vertices, one can define to be the maximum possible density of a graph that does not contain a graph as a minor. This quantity has been a subject of very intensive research; for example, a long list of bounds concerning culminated with a result of Thomason in 2001, who precisely determined its asymptotic behavior. This paper provides bounds on when itself is from a class of sparse graphs. In particular, the authors prove an asymptotically tight bound on in terms of the number of vertices of and the ratio of the vertex cover and the number of vertices of graphs contained in a class of graphs with strongly sublinear separators.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47437456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Transversal factors and spanning trees","authors":"R. Montgomery, Alp Muyesser, Yanitsa Pehova","doi":"10.19086/aic.2022.3","DOIUrl":"https://doi.org/10.19086/aic.2022.3","url":null,"abstract":"Since the proof of a \"colorful\" version of [Caratheodory's theorem](https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_%28convex_hull%29) by Bárány in 1982, it has been an important problem to obtain colorful extensions of other classical results in discrete geometry (for instance Tverberg's theorem). The present paper continues this line of research, but in the context of extremal graph theory rather than discrete geometry.\u0000\u0000Mantel's classical theorem from 1907 states that every $n$-vertex graph on more than $n^2/4$ edges contains a triangle. In [Ron Aharoni, Matt DeVos, Sebastián González Hermosillo de la Maza, Amanda Montejano, and Robert Šámal, A rainbow version of Mantel’s Theorem, Advances in Combinatorics 2020:2, 12 pp](https://arxiv.org/abs/1812.11872v2), a \"rainbow\", \"colored\", or \"colorful\" variant of this problem was considered : given three graphs $G_1,G_2,G_3$ on the same vertex set of size $n$, what average degree conditions on $G_1,G_2,G_3$ force the existence of a \"rainbow triangle\" (a triangle ${e_1,e_2,e_3}$ such that each edge $e_i$ belongs to $G_i$)? By taking three copies of the same graph $G$ we see that the colored version is at least as hard as the original problem, and the paper cited above provided a construction showing that in this case the colorful variant is strictly harder than Mantel's problem. \u0000\u0000It was suggested to study average degree or minimum degree thresholds for colorful variants of classical problems in extremal combinatorics, such as Dirac's theorem (every $n$-vertex graph of minimum degree at least $n/2$ has a Hamiltonian cycle). In particular, Joos and Kim proved in 2020 that the same minimum degree condition as in Dirac's theorem guarantees a rainbow $n$-cycle: namely if we are given $n$ graphs of minimum degree at least $n/2$ on the same set of $n$ vertices, then there is an $n$-cycle comprising one edge of each graph.\u0000\u0000The results in the present paper follow the same line of research. The two major results that are extended to the colorful setting here are a theorem of Kühn and Osthus (a sharp minimum degree condition to obtain a perfect packing of copies of any given graph $F$, generalizing the Hajnal-Szemerédi theorem), and a theorem of Komlós, Sárközy and Szemerédi (a sharp degree condition to contain any given spanning tree without large degree vertices). Amazingly, the minimum degree conditions in the (stronger) colorful versions are the same as the original minimum degree conditions.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48952588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Leaper Tours","authors":"Nikolai Beluhov","doi":"10.19086/aic.2022.4","DOIUrl":"https://doi.org/10.19086/aic.2022.4","url":null,"abstract":"Let $p$ and $q$ be positive integers. The $(p, q)$-leaper $L$ is a generalised knight which leaps $p$ units away along one coordinate axis and $q$ units away along the other. Consider a free $L$, meaning that $p + q$ is odd and $p$ and $q$ are relatively prime. We prove that $L$ tours the board of size $4pq times n$ for all sufficiently large positive integers $n$. Combining this with the recently established conjecture of Willcocks which states that $L$ tours the square board of side $2(p + q)$, we conclude that furthermore $L$ tours all boards both of whose sides are even and sufficiently large. This, in particular, completely resolves the question of the Hamiltonicity of leaper graphs on sufficiently large square boards.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43248002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cycle Lengths Modulo k in Large 3-connected Cubic Graphs, Advances in Combinatorics","authors":"K. S. Lyngsie, Martin Merker","doi":"10.19086/AIC.18971","DOIUrl":"https://doi.org/10.19086/AIC.18971","url":null,"abstract":"The existence of cycles with a given length is classical topic in graph theory with a plethora of open problems. Examples related to the main result of this paper include a conjecture of Burr and Erdős from 1976 asked whether for every integer $m$ and a positive odd integer $k$, there exists $d$ such that every graph with average degree at least $d$ contains a cycle of length $m$ modulo $k$; this conjecture was proven by Bollobás in [Bull. London Math. Soc. 9 (1977), 97-98]( https://doi.org/10.1112/blms/9.1.97). Another example is a problem of Erdős from the 1990s asking whether there exists $Asubseteqmathbb{N}$ with zero density and constants $n_0$ and $d_0$ such that every graph with at least $n_0$ vertices and the average degree at least $d_0$ contains a cycle with length in the set $A$, which was resolved by Verstraete in [J. Graph Theory 49 (2005), 151-167]( https://doi.org/10.1002/jgt.20072). In 1983, Thomassen conjectured that for all integers $m$ and $k$, every graph with minimum degree $k+1$ contains a cycle of length $2m$ modulo $k$. Note that the parity condition in the first and the third conjectures is necessary because of bipartite graphs.\u0000\u0000The current paper contributes to this long line of research by proving that for every integer $m$ and a positive odd integer $k$, every sufficiently large $3$-connected cubic graph contains a cycle of length $m$ modulo $k$. The result is the best possible in the sense that the same conclusion is not true for $2$-connected cubic graphs or $3$-connected graphs with minimum degree three.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42785387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Box and Segment Intersection Graphs with Large Girth and Chromatic Number","authors":"James Davies","doi":"10.19086/aic.25431","DOIUrl":"https://doi.org/10.19086/aic.25431","url":null,"abstract":"We prove that there are intersection graphs of axis-aligned boxes in R3 and\u0000intersection graphs of straight lines in R3 that have arbitrarily large girth and chromatic\u0000number.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42791823","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymptotic Enumeration of Hypergraphs by Degree Sequence","authors":"Nina Kamvcev, Anita Liebenau, N. Wormald","doi":"10.19086/aic.32357","DOIUrl":"https://doi.org/10.19086/aic.32357","url":null,"abstract":"We prove an asymptotic formula for the number of k-uniform hypergraphs with\u0000a given degree sequence, for a wide range of parameters. In particular, we find a formula\u0000that is asymptotically equal to the number of d-regular k-uniform hypergraphs on n vertices provided that dn ≤ c(n/k) for a constant c > 0, and 3 ≤ k < n^c for any C < 1/9. Our results relate the degree sequence of a random k-uniform hypergraph to a simple model of nearly independent binomial random variables, thus extending the recent results for graphs due to the second and third author.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48941315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exact stability for Turán’s Theorem","authors":"D'aniel Kor'andi, Alexander Roberts, A. Scott","doi":"10.19086/aic.31079","DOIUrl":"https://doi.org/10.19086/aic.31079","url":null,"abstract":"Turán's Theorem says that an extremal Kr+1-free graph is r-partite. The Stability Theorem of Erdős and Simonovits shows that if a Kr+1-free graph with n vertices has close to the maximal tr(n) edges, then it is close to being r-partite. In this paper we determine exactly the Kr+1-free graphs with at least m edges that are farthest from being r-partite, for any m≥tr(n)−δrn2. This extends work by Erdős, Győri and Simonovits, and proves a conjecture of Balogh, Clemen, Lavrov, Lidický and Pfender.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41759472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The size-Ramsey number of 3-uniform tight paths","authors":"Jie Han, Y. Kohayakawa, Shoham Letzter, G. Mota, Olaf Parczyk","doi":"10.19086/aic.24581","DOIUrl":"https://doi.org/10.19086/aic.24581","url":null,"abstract":"Given a hypergraph H, the size-Ramsey number r(H) is the smallest integer m such that there exists a graph G with m edges with the property that in any colouring of the edges of G with two colours there is amonochromatic copy of H. We prove that the size-Ramsey number of the 3-uniform tight path on n vertices P_n is linear in n, i.e., r(P_n)=O(n). This answers a question by Dudek, Fleur, Mubayi, and Rödl for 3-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved r(P_n)=O(n^1.5*log^1.5 n).","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49051669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A rainbow version of Mantel's Theorem","authors":"Robert Šámal, A. Montejano, Sebastián González Hermosillo de la Maza, Matt DeVos, R. Aharoni","doi":"10.19086/aic.12043","DOIUrl":"https://doi.org/10.19086/aic.12043","url":null,"abstract":"Mantel's Theorem from 1907 is one of the oldest results in graph theory: every simple $n$-vertex graph with more than $frac{1}{4}n^2$ edges contains a triangle. The theorem has been generalized in many different ways, including other subgraphs, minimum degree conditions, etc. This article deals with a generalization to edge-colored multigraphs, which can be viewed as a union of simple graphs, each corresponding to an edge-color class.\u0000\u0000The case of two colors is the same as the original setting: Diwan and Mubayi proved that any two graphs $G_1$ and $G_2$ on the same set of $n$ vertices, each containing more than $frac{1}{4}n^2$ edges, give rise to a triangle with one edge from $G_1$ and two edges from $G_2$. The situation is however different for three colors. Fix $tau=frac{4-sqrt{7}}{9}$ and split the $n$ vertices into three sets $A$, $B$ and $C$, such that $|B|=|C|=lfloortau nrfloor$ and $|A|=n-|B|-|C|$. The graph $G_1$ contains all edges inside $A$ and inside $B$, the graph $G_2$ contains all edges inside $A$ and inside $C$, and the graph $G_3$ contains all edges between $A$ and $Bcup C$ and inside $Bcup C$. It is easy to check that there is no triangle with one edge from $G_1$, one from $G_2$ and one from $G_3$; each of the graphs has about $frac{1+tau^2}{4}n^2=frac{26-2sqrt{7}}{81}n^2approx 0.25566n^2$ edges. The main result of the article is that this construction is optimal: any three graphs $G_1$, $G_2$ and $G_3$ on the same set of $n$ vertices, each containing more than $frac{1+tau^2}{4}n^2$ edges, give rise to a triangle with one edge from each of the graphs $G_1$, $G_2$ and $G_3$. A computer-assisted proof of the same bound has been found by Culver, Lidický, Pfender and Volec.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49137151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The number of partial Steiner systems and d-partitions","authors":"R. Hofstad, R. Pendavingh, J. V. D. Pol","doi":"10.19086/aic.32563","DOIUrl":"https://doi.org/10.19086/aic.32563","url":null,"abstract":"We prove asymptotic upper bounds on the number of $d$-partitions (paving matroids of fixed rank) and partial Steiner systems (sparse paving matroids of fixed rank), using a mixture of entropy counting, sparse encoding, and the probabilistic method.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43497010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}