DeVos, Matt, Funk, Daryl, Goddyn, Luis, Royle, Gordon
We show that the class of bicircular matroids has only a finite number of excluded minors. Key tools used in our proof include representations of matroids by biased graphs and the recently introduced class of quasi-graphic matroids. We show that if $N$ is an excluded minor of rank at least ten, then $N$ is quasi-graphic. Several small excluded minors are quasi-graphic. Using biased-graphic representations, we find that $N$ already contains one of these. We also provide an upper bound, in terms of rank, on the number of elements in an excluded minor, so the result follows.
{"title":"There are only a finite number of excluded minors for the class of bicircular matroids","authors":"DeVos, Matt, Funk, Daryl, Goddyn, Luis, Royle, Gordon","doi":"10.19086/aic.2023.7","DOIUrl":"https://doi.org/10.19086/aic.2023.7","url":null,"abstract":"We show that the class of bicircular matroids has only a finite number of excluded minors. Key tools used in our proof include representations of matroids by biased graphs and the recently introduced class of quasi-graphic matroids. We show that if $N$ is an excluded minor of rank at least ten, then $N$ is quasi-graphic. Several small excluded minors are quasi-graphic. Using biased-graphic representations, we find that $N$ already contains one of these. We also provide an upper bound, in terms of rank, on the number of elements in an excluded minor, so the result follows.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136227530","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}
Bonamy, Marthe, Botler, Fábio, Dross, François, Naia, Tássio, Skokan, Jozef
Recently, Letzter proved that any graph of order $n$ contains a collection $mathcal{P}$ of $O(nlog^star n)$ paths with the following property: for all distinct edges $e$ and $f$ there exists a path in $mathcal{P}$ which contains $e$ but not $f$. We improve this upper bound to $19 n$, thus answering a question of G.O.H. Katona and confirming a conjecture independently posed by Balogh, Csaba, Martin, and Pluh'ar and by Falgas-Ravry, Kittipassorn, Kor'andi, Letzter, and Narayanan. Our proof is elementary and self-contained.
{"title":"Separating the edges of a graph by a linear number of paths","authors":"Bonamy, Marthe, Botler, Fábio, Dross, François, Naia, Tássio, Skokan, Jozef","doi":"10.19086/aic.2023.6","DOIUrl":"https://doi.org/10.19086/aic.2023.6","url":null,"abstract":"Recently, Letzter proved that any graph of order $n$ contains a collection $mathcal{P}$ of $O(nlog^star n)$ paths with the following property: for all distinct edges $e$ and $f$ there exists a path in $mathcal{P}$ which contains $e$ but not $f$. We improve this upper bound to $19 n$, thus answering a question of G.O.H. Katona and confirming a conjecture independently posed by Balogh, Csaba, Martin, and Pluh'ar and by Falgas-Ravry, Kittipassorn, Kor'andi, Letzter, and Narayanan. Our proof is elementary and self-contained.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234355","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}
Extending an earlier conjecture of ErdH{o}s, Burr and Rosta conjectured that among all two-colorings of the edges of a complete graph, the uniformly random coloring asymptotically minimizes the number of monochromatic copies of any fixed graph $H$. This conjecture was disproved independently by Sidorenko and Thomason. The first author later found quantitatively stronger counterexamples, using the Tur'an coloring, in which one of the two colors spans a balanced complete multipartite graph. We prove that the Tur'an coloring is extremal for an infinite family of graphs, and that it is the unique extremal coloring. This yields the first determination of the Ramsey multiplicity constant of a graph for which the Burr--Rosta conjecture fails. We also prove an analogous three-color result. In this case, our result is conditional on a certain natural conjecture on the behavior of two-color Ramsey numbers.
{"title":"Ramsey multiplicity and the Turán coloring","authors":"J. Fox, Yuval Wigderson","doi":"10.19086/aic.2023.2","DOIUrl":"https://doi.org/10.19086/aic.2023.2","url":null,"abstract":"Extending an earlier conjecture of ErdH{o}s, Burr and Rosta conjectured that among all two-colorings of the edges of a complete graph, the uniformly random coloring asymptotically minimizes the number of monochromatic copies of any fixed graph $H$. This conjecture was disproved independently by Sidorenko and Thomason. The first author later found quantitatively stronger counterexamples, using the Tur'an coloring, in which one of the two colors spans a balanced complete multipartite graph. We prove that the Tur'an coloring is extremal for an infinite family of graphs, and that it is the unique extremal coloring. This yields the first determination of the Ramsey multiplicity constant of a graph for which the Burr--Rosta conjecture fails. We also prove an analogous three-color result. In this case, our result is conditional on a certain natural conjecture on the behavior of two-color Ramsey numbers.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42863724","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}
A longstanding conjecture of Erd˝os and Simonovits states that for every rational r between 1 and 2 there is a graph H such that the largest number of edges in an H-free graph on n vertices is Q(nr). Answering a question raised by Jiang, Jiang and Ma, we show that the conjecture holds for all rationals of the form 2a=b with b sufficiently large in terms of a.
{"title":"Rational Exponents Near Two","authors":"D. Conlon, Oliver Janzer","doi":"10.19086/aic.2022.9","DOIUrl":"https://doi.org/10.19086/aic.2022.9","url":null,"abstract":"A longstanding conjecture of Erd˝os and Simonovits states that for every rational r between 1 and 2 there is a graph H such that the largest number of edges in an H-free graph on n vertices is Q(nr). Answering a question raised by Jiang, Jiang and Ma, we show that the conjecture holds for all rationals of the form 2a=b with b sufficiently large in terms of a.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47506951","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}
Mat'ias Pavez-Sign'e, Nicolás Sanhueza-Matamala, M. Stein
We prove that for fixed $rge kge 2$, every $k$-uniform hypergraph on $n$ vertices having minimum codegree at least $(1-(binom{r-1}{k-1}+binom{r-2}{k-2})^{-1})n+o(n)$ contains the $(r-k+1)$th power of a tight Hamilton cycle. This result may be seen as a step towards a hypergraph version of the P'osa-Seymour conjecture. Moreover, we prove that the same bound on the codegree suffices for finding a copy of every spanning hypergraph of tree-width less than $r$ which admits a tree decomposition where every vertex is in a bounded number of bags.
{"title":"Towards a Hypergraph version of the Pósa–Seymour Conjecture","authors":"Mat'ias Pavez-Sign'e, Nicolás Sanhueza-Matamala, M. Stein","doi":"10.19086/aic.2023.3","DOIUrl":"https://doi.org/10.19086/aic.2023.3","url":null,"abstract":"We prove that for fixed $rge kge 2$, every $k$-uniform hypergraph on $n$ vertices having minimum codegree at least $(1-(binom{r-1}{k-1}+binom{r-2}{k-2})^{-1})n+o(n)$ contains the $(r-k+1)$th power of a tight Hamilton cycle. This result may be seen as a step towards a hypergraph version of the P'osa-Seymour conjecture. Moreover, we prove that the same bound on the codegree suffices for finding a copy of every spanning hypergraph of tree-width less than $r$ which admits a tree decomposition where every vertex is in a bounded number of bags.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49116249","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 Ramsey number $r(G,H)$ is the minimum $N$ such that every graph on $N$ vertices contains $G$ as a subgraph or its complement contains $H$ as a subgraph. For integers $n geq k geq 1$, the $k$-book $B_{k,n}$ is the graph on $n$ vertices consisting of a copy of $K_k$, called the spine, as well as $n-k$ additional vertices each adjacent to every vertex of the spine and non-adjacent to each other. A connected graph $H$ on $n$ vertices is called $p$-good if $r(K_p,H)=(p-1)(n-1)+1$. Nikiforov and Rousseau proved that if $n$ is sufficiently large in terms of $p$ and $k$, then $B_{k,n}$ is $p$-good. Their proof uses Szemer'edi's regularity lemma and gives a tower-type bound on $n$. We give a short new proof that avoids using the regularity method and shows that every $B_{k,n}$ with $n geq 2^{k^{10p}}$ is $p$-good. Using Szemer'edi's regularity lemma, Nikiforov and Rousseau also proved much more general goodness-type results, proving a tight bound on $r(G,H)$ for several families of sparse graphs $G$ and $H$ as long as $|V(G)|0$. Using our techniques, we prove a new result of this type, showing that $r(G,H) = (p-1)(n-1)+1$ when $H =B_{k,n}$ and $G$ is a complete $p$-partite graph whose first $p-1$ parts have constant size and whose last part has size $delta n$, for some small constant $delta>0$. Again, our proof does not use the regularity method, and thus yields double-exponential bounds on $delta$.
{"title":"Ramsey goodness of books revisited","authors":"J. Fox, Xiaoyu He, Yuval Wigderson","doi":"10.19086/aic.2023.4","DOIUrl":"https://doi.org/10.19086/aic.2023.4","url":null,"abstract":"The Ramsey number $r(G,H)$ is the minimum $N$ such that every graph on $N$ vertices contains $G$ as a subgraph or its complement contains $H$ as a subgraph. For integers $n geq k geq 1$, the $k$-book $B_{k,n}$ is the graph on $n$ vertices consisting of a copy of $K_k$, called the spine, as well as $n-k$ additional vertices each adjacent to every vertex of the spine and non-adjacent to each other. A connected graph $H$ on $n$ vertices is called $p$-good if $r(K_p,H)=(p-1)(n-1)+1$. Nikiforov and Rousseau proved that if $n$ is sufficiently large in terms of $p$ and $k$, then $B_{k,n}$ is $p$-good. Their proof uses Szemer'edi's regularity lemma and gives a tower-type bound on $n$. We give a short new proof that avoids using the regularity method and shows that every $B_{k,n}$ with $n geq 2^{k^{10p}}$ is $p$-good. Using Szemer'edi's regularity lemma, Nikiforov and Rousseau also proved much more general goodness-type results, proving a tight bound on $r(G,H)$ for several families of sparse graphs $G$ and $H$ as long as $|V(G)|<delta |V(H)|$ for a small constant $delta>0$. Using our techniques, we prove a new result of this type, showing that $r(G,H) = (p-1)(n-1)+1$ when $H =B_{k,n}$ and $G$ is a complete $p$-partite graph whose first $p-1$ parts have constant size and whose last part has size $delta n$, for some small constant $delta>0$. Again, our proof does not use the regularity method, and thus yields double-exponential bounds on $delta$.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44849840","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}
Tara Abrishami, M. Chudnovsky, Sepehr Hajebi, S. Spirkl
A _theta_ is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $mathcal{H}$ of graphs, we say a graph $G$ is $mathcal{H}$-_free_ if no induced subgraph of $G$ is isomorphic to a member of $mathcal{H}$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $c$ for which every (theta, triangle)-free graph $G$ has treewidth at most $clog (|V(G)|)$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth. Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in $|V(G)|$ for every graph $G$ excluding the so-called _three-path-configurations_ as well as a fixed complete graph. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and $k$-Coloring (for fixed $k$) admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph.
{"title":"Induced subgraphs and tree decompositions III. Three-path-configurations and logarithmic treewidth","authors":"Tara Abrishami, M. Chudnovsky, Sepehr Hajebi, S. Spirkl","doi":"10.19086/aic.2022.6","DOIUrl":"https://doi.org/10.19086/aic.2022.6","url":null,"abstract":"A _theta_ is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $mathcal{H}$ of graphs, we say a graph $G$ is $mathcal{H}$-_free_ if no induced subgraph of $G$ is isomorphic to a member of $mathcal{H}$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $c$ for which every (theta, triangle)-free graph $G$ has treewidth at most $clog (|V(G)|)$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth.\u0000\u0000Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in $|V(G)|$ for every graph $G$ excluding the so-called _three-path-configurations_ as well as a fixed complete graph. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and $k$-Coloring (for fixed $k$) admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47041693","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 Ramsey number r(G) of a graph G is the smallest integer n such that any 2 colouring of the edges of a clique on n vertices contains a monochromatic copy of G. Determining the Ramsey number of G is a central problem of Ramsey theory with long and illustrious history. Despite this there are precious few classes of graphs G for which the value of r(G) is known exactly. One such family consists of large vertex disjoint unions of a fixed graph H, we denote such a graph, consisting of n copies of H by nH. This classical result was proved by Burr, Erd˝os and Spencer in 1975, who showed r(nH) = (2jHja(H))n+c, for some c = c(H), provided n is large enough. Since it did not follow from their arguments, Burr, Erd˝os and Spencer further asked to determine the number of copies we need to take in order to see this long term behaviour and the value of c. More than 30 years ago Burr gave a way of determining c(H), which only applies when the number of copies n is triple exponential in jHj. In this paper we give an essentially tight answer to this very old problem of Burr, Erd˝os and Spencer by showing that the long term behaviour occurs already when the number of copies is single exponential.
{"title":"Tight Ramsey Bounds for Multiple Copies of a Graph","authors":"Matija Bucić, B. Sudakov","doi":"10.19086/aic.2023.1","DOIUrl":"https://doi.org/10.19086/aic.2023.1","url":null,"abstract":"The Ramsey number r(G) of a graph G is the smallest integer n such that any\u00002 colouring of the edges of a clique on n vertices contains a monochromatic copy of G. Determining the Ramsey number of G is a central problem of Ramsey theory with long and illustrious history. Despite this there are precious few classes of graphs G for which the value of r(G) is known exactly. One such family consists of large vertex disjoint unions of a fixed\u0000graph H, we denote such a graph, consisting of n copies of H by nH. This classical result was proved by Burr, Erd˝os and Spencer in 1975, who showed r(nH) = (2jHja(H))n+c, for some c = c(H), provided n is large enough. Since it did not follow from their arguments, Burr, Erd˝os and Spencer further asked to determine the number of copies we need to take\u0000in order to see this long term behaviour and the value of c. More than 30 years ago Burr gave a way of determining c(H), which only applies when the number of copies n is triple exponential in jHj. In this paper we give an essentially tight answer to this very old problem of Burr, Erd˝os and Spencer by showing that the long term behaviour occurs already when the number of copies is single exponential.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42769799","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}
In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erd˝os, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of girth at least five is 3-choosable. In this paper, we introduce the concept of a local girth list assignment: a list assignment wherein the list size of a vertex depends not on the girth of the graph, but rather on the length of the shortest cycle in which the vertex is contained. We give a local list colouring theorem unifying the two theorems of Thomassen mentioned above. In particular, we show that if G is a planar graph and L is a list assignment for G such that |L(v)| ≥ 3 for all v ∈ V(G); |L(v)| ≥ 4 for every vertex v contained in a 4-cycle; and |L(v)| ≥ 5 for every v contained in a triangle, then G admits an L-colouring.
{"title":"Local Girth Choosability of Planar Graphs","authors":"Luke Postle, Evelyne Smith-Roberge","doi":"10.19086/aic.2022.8","DOIUrl":"https://doi.org/10.19086/aic.2022.8","url":null,"abstract":"In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erd˝os, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of girth at least five is 3-choosable. In this paper, we introduce the concept of a local girth list assignment: a list assignment wherein the list size of a vertex depends not on the girth of the graph, but rather on the length of the shortest cycle in which the vertex is contained. We give a local list colouring theorem unifying the two theorems of Thomassen mentioned above. In particular, we show that if G is a planar graph and L is a list assignment for G such that |L(v)| ≥ 3 for all v ∈ V(G); |L(v)| ≥ 4 for every vertex v contained in a 4-cycle; and |L(v)| ≥ 5 for every v contained in a triangle, then G admits an L-colouring.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43208297","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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