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. The 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). As 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.
{"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}
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. Mantel'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. It 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. The 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.
自从1982年Bárány证明了[卡拉多里定理](https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_%28convex_hull%29)的“彩色”版本以来，在离散几何中获得其他经典结果的彩色扩展(例如特弗伯格定理)一直是一个重要的问题。本文继续这条研究路线，但在极值图论的背景下，而不是离散几何。曼特尔在1907年提出的经典定理指出，在超过n^2/4$条边上，每$n$顶点的图都包含一个三角形。在[Ron Aharoni, Matt DeVos, Sebastián González Hermosillo de la Maza, Amanda Montejano, and Robert Šámal，彩虹版的Mantel定理，Advances In Combinatorics 2020:2, 12 pp](https://arxiv.org/abs/1812.11872v2)中，考虑了这个问题的“彩虹”，“彩色”或“彩色”变体:给定三个图$G_1,G_2,G_3$在相同大小为$n$的顶点集上，$G_1,G_2,G_3$上的平均度数条件是什么迫使“彩虹三角形”(一个三角形${e_1,e_2,e_3}$使得每条边$e_i$都属于$G_i$)的存在?通过取同一图$G$的三个副本，我们看到彩色版本至少和原始问题一样难，上面引用的论文提供了一个结构，表明在这种情况下，彩色版本严格地比Mantel的问题更难。建议研究极值组合中经典问题的彩色变体的平均度或最小度阈值，如狄拉克定理(至少$n/2$的最小度的$n$顶点图有一个哈密顿循环)。特别是，Joos和Kim在2020年证明了与Dirac定理中相同的最小度条件保证了彩虹$n$-环:即如果我们在相同的$n$顶点集上给定$n$最小度至少$n/2$的图，则存在包含每个图的一条边的$n$-环。本文的研究结果遵循了相同的研究思路。扩展到这里的两个主要结果是k hn和Osthus定理(获得任何给定图$F$副本的完美包合的锐最小度条件，推广hajnal - szemersamedi定理)，以及Komlós, Sárközy和szemersamedi定理(包含任何给定的无大度顶点的生成树的锐度条件)。令人惊讶的是，(较强)彩色版本中的最小度数条件与原始的最小度数条件相同。
{"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}
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.
{"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}
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. The 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.
给定长度的环的存在性是图论中的经典话题，有很多开放问题。与本文主要结果有关的例子包括Burr和Erdös从1976年提出的一个猜想，即对于每一个整数$m$和一个正奇整数$k$，是否存在$d$，使得每个平均度至少为$d$的图都包含一个长度为$m$模$k$的循环；Bollobás在[Bull.London Math.Soc.9（1977），97-98]（https://doi.org/10.1112/blms/9.1.97)。另一个例子是20世纪90年代Erdös提出的一个问题，即是否存在密度为零、常数为$N_0$和$d_0$的$asubsteqmathbb｛N｝$，使得每个顶点至少为$N_0$、平均度至少为$d_0$的图都包含一个长度在集合$a$中的循环，Verstraete在[J.graph Theory 49（2005），151-167]（https://doi.org/10.1002/jgt.20072)。1983年，托马森推测，对于所有整数$m$和$k$，每个具有最小度$k+1$的图都包含一个长度为$2m$模$k$的循环。注意，由于二分图，第一和第三猜想中的奇偶条件是必要的。本文通过证明对于每一个整数$m$和一个正奇整数$k$，每一个足够大的$3$连通三次图都包含一个长度为$m$模$k$的循环，为这一长期研究做出了贡献。该结果是最好的，因为对于最小阶为3的$2$连通三次图或$3$连通图，相同的结论不成立。
{"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}
We prove that there are intersection graphs of axis-aligned boxes in R3 and intersection graphs of straight lines in R3 that have arbitrarily large girth and chromatic number.
证明了在R3中存在轴向方框的交图和具有任意大周长和色数的直线的交图。
{"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}
We prove an asymptotic formula for the number of k-uniform hypergraphs with a given degree sequence, for a wide range of parameters. In particular, we find a formula that 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.
{"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}
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.
{"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}
Jie Han, Y. Kohayakawa, Shoham Letzter, G. Mota, Olaf Parczyk
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).
给定超图H，Ramsey数r（H）是最小整数m，使得存在一个具有m条边的图G，其性质是在具有两种颜色的G的边的任何着色中都存在H的单色副本。我们证明了n个顶点P_n上的3-均匀紧路径的Ramsey数在n中是线性的，即r（P_n）=O（n）。这回答了Dudek、Fleur、Mubayi和Rödl关于3-一致超图的一个问题[关于超图的大小Ramsey数，J.Graph Theory 86（2016），417-434]，他证明了R（P_n）=O（n^1.5*log^1.5n）。
{"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}
Robert Šámal, A. Montejano, Sebastián González Hermosillo de la Maza, Matt DeVos, R. Aharoni
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. The 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.
{"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}
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.
{"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}
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