Pub Date : 2018-07-16DOI: 10.4310/joc.2021.v12.n2.a4
Z. Furedi, A. Kostochka, Ruth Luo
Let $EG_r(n,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph with no Berge cycles of length $k$ or longer. In the first part of this work, we have found exact values of $EG_r(n,k)$ and described the structure of extremal hypergraphs for the case when $k-2$ divides $n-1$ and $kgeq r+3$. In this paper we determine $EG_r(n,k)$ and describe the extremal hypergraphs for all $n$ when $kgeq r+4$.
{"title":"Avoiding long Berge cycles II, exact bounds for all $n$","authors":"Z. Furedi, A. Kostochka, Ruth Luo","doi":"10.4310/joc.2021.v12.n2.a4","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n2.a4","url":null,"abstract":"Let $EG_r(n,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph with no Berge cycles of length $k$ or longer. In the first part of this work, we have found exact values of $EG_r(n,k)$ and described the structure of extremal hypergraphs for the case when $k-2$ divides $n-1$ and $kgeq r+3$. In this paper we determine $EG_r(n,k)$ and describe the extremal hypergraphs for all $n$ when $kgeq r+4$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"58 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84445709","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}
Pub Date : 2018-06-22DOI: 10.4310/joc.2022.v13.n1.a1
Thomas Schweser, M. Stiebitz, B. Toft
For a hypergraph $G$, let $chi(G), Delta(G),$ and $lambda(G)$ denote the chromatic number, the maximum degree, and the maximum local edge connectivity of $G$, respectively. A result of Rhys Price Jones from 1975 says that every connected hypergraph $G$ satisfies $chi(G) leq Delta(G) + 1$ and equality holds if and only if $G$ is a complete graph, an odd cycle, or $G$ has just one (hyper-)edge. By a result of Bjarne Toft from 1970 it follows that every hypergraph $G$ satisfies $chi(G) leq lambda(G) + 1$. In this paper, we show that a hypergraph $G$ with $lambda(G) geq 3$ satisfies $chi(G) = lambda(G) + 1$ if and only if $G$ contains a block which belongs to a family $mathcal{H}_{lambda(G)}$. The class $mathcal{H}_3$ is the smallest family which contains all odd wheels and is closed under taking Haj'os joins. For $k geq 4$, the family $mathcal{H}_k$ is the smallest that contains all complete graphs $K_{k+1}$ and is closed under Haj'os joins. For the proofs of the above results we use critical hypergraphs. A hypergraph $G$ is called $(k+1)$-critical if $chi(G)=k+1$, but $chi(H)leq k$ whenever $H$ is a proper subhypergraph of $G$. We give a characterization of $(k+1)$-critical hypergraphs having a separating edge set of size $k$ as well as a a characterization of $(k+1)$-critical hypergraphs having a separating vertex set of size $2$.
{"title":"Coloring hypergraphs of low connectivity","authors":"Thomas Schweser, M. Stiebitz, B. Toft","doi":"10.4310/joc.2022.v13.n1.a1","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n1.a1","url":null,"abstract":"For a hypergraph $G$, let $chi(G), Delta(G),$ and $lambda(G)$ denote the chromatic number, the maximum degree, and the maximum local edge connectivity of $G$, respectively. A result of Rhys Price Jones from 1975 says that every connected hypergraph $G$ satisfies $chi(G) leq Delta(G) + 1$ and equality holds if and only if $G$ is a complete graph, an odd cycle, or $G$ has just one (hyper-)edge. By a result of Bjarne Toft from 1970 it follows that every hypergraph $G$ satisfies $chi(G) leq lambda(G) + 1$. In this paper, we show that a hypergraph $G$ with $lambda(G) geq 3$ satisfies $chi(G) = lambda(G) + 1$ if and only if $G$ contains a block which belongs to a family $mathcal{H}_{lambda(G)}$. The class $mathcal{H}_3$ is the smallest family which contains all odd wheels and is closed under taking Haj'os joins. For $k geq 4$, the family $mathcal{H}_k$ is the smallest that contains all complete graphs $K_{k+1}$ and is closed under Haj'os joins. For the proofs of the above results we use critical hypergraphs. A hypergraph $G$ is called $(k+1)$-critical if $chi(G)=k+1$, but $chi(H)leq k$ whenever $H$ is a proper subhypergraph of $G$. We give a characterization of $(k+1)$-critical hypergraphs having a separating edge set of size $k$ as well as a a characterization of $(k+1)$-critical hypergraphs having a separating vertex set of size $2$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"100 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85841709","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}
Pub Date : 2018-06-04DOI: 10.4310/joc.2020.v11.n4.a6
E. Gunawan, R. Schiffler
Let Q be a quiver without loops and 2-cycles, let A(Q) be the corresponding cluster algebra and let x be a cluster. We introduce a new class of integer vectors which we call frieze vectors relative to x. These frieze vectors are defined as solutions of certain Diophantine equations given by the cluster variables in the cluster algebra. We show that every cluster gives rise to a frieze vector and that the frieze vector determines the cluster. We also study friezes of type Q as homomorphisms from the cluster algebra to an arbitrary integral domain. In particular, we show that every positive integral frieze of affine Dynkin type A is unitary, which means it is obtained by specializing each cluster variable in one cluster to the constant 1. This completes the answer to the question of unitarity for all positive integral friezes of Dynkin and affine Dynkin types.
{"title":"Frieze vectors and unitary friezes","authors":"E. Gunawan, R. Schiffler","doi":"10.4310/joc.2020.v11.n4.a6","DOIUrl":"https://doi.org/10.4310/joc.2020.v11.n4.a6","url":null,"abstract":"Let Q be a quiver without loops and 2-cycles, let A(Q) be the corresponding cluster algebra and let x be a cluster. We introduce a new class of integer vectors which we call frieze vectors relative to x. These frieze vectors are defined as solutions of certain Diophantine equations given by the cluster variables in the cluster algebra. We show that every cluster gives rise to a frieze vector and that the frieze vector determines the cluster. \u0000We also study friezes of type Q as homomorphisms from the cluster algebra to an arbitrary integral domain. In particular, we show that every positive integral frieze of affine Dynkin type A is unitary, which means it is obtained by specializing each cluster variable in one cluster to the constant 1. This completes the answer to the question of unitarity for all positive integral friezes of Dynkin and affine Dynkin types.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"73 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90586516","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}
Pub Date : 2018-05-11DOI: 10.4310/JOC.2019.V10.N1.A7
H. Fleischner, G. Chia
This is the second part of joint research in which we show that every $2$-connected graph $G$ has the ${cal F}_4$ property. That is, given distinct $x_iin V(G)$, $1leq ileq 4$, there is an $x_1x_2$-hamiltonian path in $G^2$ containing different edges $x_3y_3, x_4y_4in E(G)$ for some $y_3,y_4in V(G)$. However, it was shown already in cite[Theorem 2]{cf1:refer} that 2-connected DT-graphs have the ${cal F}_4$ property; based on this result we generalize it to arbitrary $2$-connected graphs. We also show that these results are best possible.
{"title":"Revisiting the Hamiltonian theme in the square of a block: the general case","authors":"H. Fleischner, G. Chia","doi":"10.4310/JOC.2019.V10.N1.A7","DOIUrl":"https://doi.org/10.4310/JOC.2019.V10.N1.A7","url":null,"abstract":"This is the second part of joint research in which we show that every $2$-connected graph $G$ has the ${cal F}_4$ property. That is, given distinct $x_iin V(G)$, $1leq ileq 4$, there is an $x_1x_2$-hamiltonian path in $G^2$ containing different edges $x_3y_3, x_4y_4in E(G)$ for some $y_3,y_4in V(G)$. However, it was shown already in cite[Theorem 2]{cf1:refer} that 2-connected DT-graphs have the ${cal F}_4$ property; based on this result we generalize it to arbitrary $2$-connected graphs. We also show that these results are best possible.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"26 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74224650","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}
Pub Date : 2018-05-10DOI: 10.4310/joc.2021.v12.n1.a6
Chang-Pao Chen, Catherine S. Greenhill
The study of substructures in random objects has a long history, beginning with Erdős and Renyi's work on subgraphs of random graphs. We study the existence of certain substructures in random subsets of vector spaces over finite fields. First we provide a general framework which can be applied to establish coarse threshold results and prove a limiting Poisson distribution at the threshold scale. To illustrate our framework we apply our results to $k$-term arithmetic progressions, sums, right triangles, parallelograms and affine planes. We also find coarse thresholds for the property that a random subset of a finite vector space is sum-free, or is a Sidon set.
{"title":"Threshold functions for substructures in random subsets of finite vector spaces","authors":"Chang-Pao Chen, Catherine S. Greenhill","doi":"10.4310/joc.2021.v12.n1.a6","DOIUrl":"https://doi.org/10.4310/joc.2021.v12.n1.a6","url":null,"abstract":"The study of substructures in random objects has a long history, beginning with Erdős and Renyi's work on subgraphs of random graphs. We study the existence of certain substructures in random subsets of vector spaces over finite fields. First we provide a general framework which can be applied to establish coarse threshold results and prove a limiting Poisson distribution at the threshold scale. To illustrate our framework we apply our results to $k$-term arithmetic progressions, sums, right triangles, parallelograms and affine planes. We also find coarse thresholds for the property that a random subset of a finite vector space is sum-free, or is a Sidon set.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"46 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87541873","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}
Pub Date : 2018-05-10DOI: 10.4310/JOC.2019.V10.N4.A6
Cesar Ceballos, Rafael S. Gonz'alez D'Le'on
The Catalan numbers constitute one of the most important sequences in combinatorics. Catalan objects have been generalized in various directions, including the classical Fuss-Catalan objects and the rational Catalan generalization of Armstrong-Rhoades-Williams. We propose a wider generalization of these families indexed by a composition $s$ which is motivated by the combinatorics of planar rooted trees; when $s=(2,...,2)$ and $s=(k+1,...,k+1)$ we recover the classical Catalan and Fuss-Catalan combinatorics, respectively. Furthermore, to each pair $(a,b)$ of relatively prime numbers we can associate a signature that recovers the combinatorics of rational Catalan objects. We present explicit bijections between the resulting $s$-Catalan objects, and a fundamental recurrence that generalizes the fundamental recurrence of the classical Catalan numbers. Our framework allows us to define signature generalizations of parking functions which coincide with the generalized parking functions studied by Pitman-Stanley and Yan, as well as generalizations of permutations which coincide with the notion of Stirling multipermutations introduced by Gessel-Stanley. Some of our constructions differ from the ones of Armstrong-Rhoades-Williams, however as a byproduct of our extension, we obtain the additional notions of rational permutations and rational trees.
{"title":"Signature Catalan combinatorics","authors":"Cesar Ceballos, Rafael S. Gonz'alez D'Le'on","doi":"10.4310/JOC.2019.V10.N4.A6","DOIUrl":"https://doi.org/10.4310/JOC.2019.V10.N4.A6","url":null,"abstract":"The Catalan numbers constitute one of the most important sequences in combinatorics. Catalan objects have been generalized in various directions, including the classical Fuss-Catalan objects and the rational Catalan generalization of Armstrong-Rhoades-Williams. We propose a wider generalization of these families indexed by a composition $s$ which is motivated by the combinatorics of planar rooted trees; when $s=(2,...,2)$ and $s=(k+1,...,k+1)$ we recover the classical Catalan and Fuss-Catalan combinatorics, respectively. Furthermore, to each pair $(a,b)$ of relatively prime numbers we can associate a signature that recovers the combinatorics of rational Catalan objects. We present explicit bijections between the resulting $s$-Catalan objects, and a fundamental recurrence that generalizes the fundamental recurrence of the classical Catalan numbers. Our framework allows us to define signature generalizations of parking functions which coincide with the generalized parking functions studied by Pitman-Stanley and Yan, as well as generalizations of permutations which coincide with the notion of Stirling multipermutations introduced by Gessel-Stanley. Some of our constructions differ from the ones of Armstrong-Rhoades-Williams, however as a byproduct of our extension, we obtain the additional notions of rational permutations and rational trees.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"67 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82943475","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}
Pub Date : 2018-05-02DOI: 10.4310/joc.2019.v10.n3.a5
H. Thomas, N. Williams
Let $G$ be an acylic directed graph. For each vertex $g in G$, we define an involution on the independent sets of $G$. We call these involutions flips, and use them to define a new partial order on independent sets of $G$. Trim lattices generalize distributive lattices by removing the graded hypothesis: a graded trim lattice is a distributive lattice, and every distributive lattice is trim. Our independence posets are a further generalization of distributive lattices, eliminating also the lattice requirement: an independence poset that is a lattice is always a trim lattice, and every trim lattice is the independence poset for a unique (up to isomorphism) acyclic directed graph $G$. We characterize when an independence poset is a lattice with a graph-theoretic condition on $G$. We generalize the definition of rowmotion from distributive lattices to independence posets, and we show it can be computed in three different ways. We also relate our constructions to torsion classes, semibricks, and 2-simpleminded collections arising in the representation theory of certain acyclic finite-dimensional algebras.
{"title":"Independence posets","authors":"H. Thomas, N. Williams","doi":"10.4310/joc.2019.v10.n3.a5","DOIUrl":"https://doi.org/10.4310/joc.2019.v10.n3.a5","url":null,"abstract":"Let $G$ be an acylic directed graph. For each vertex $g in G$, we define an involution on the independent sets of $G$. We call these involutions flips, and use them to define a new partial order on independent sets of $G$. Trim lattices generalize distributive lattices by removing the graded hypothesis: a graded trim lattice is a distributive lattice, and every distributive lattice is trim. Our independence posets are a further generalization of distributive lattices, eliminating also the lattice requirement: an independence poset that is a lattice is always a trim lattice, and every trim lattice is the independence poset for a unique (up to isomorphism) acyclic directed graph $G$. We characterize when an independence poset is a lattice with a graph-theoretic condition on $G$. We generalize the definition of rowmotion from distributive lattices to independence posets, and we show it can be computed in three different ways. We also relate our constructions to torsion classes, semibricks, and 2-simpleminded collections arising in the representation theory of certain acyclic finite-dimensional algebras.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"229 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76100619","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}
Pub Date : 2018-04-26DOI: 10.4310/joc.2020.v11.n2.a2
Svante Linusson, Samu Potka
Edelman and Greene constructed a correspondence between reduced words of the reverse permutation and standard Young tableaux. We prove that for any reduced word the shape of the region of the insertion tableau containing the smallest possible entries evolves exactly as the upper-left component of the permutation's (Rothe) diagram. Properties of the Edelman-Greene bijection restricted to 132-avoiding and 2143-avoiding permutations are presented. We also consider the Edelman-Greene bijection applied to non-reduced words.
{"title":"Properties of the Edelman–Greene bijection","authors":"Svante Linusson, Samu Potka","doi":"10.4310/joc.2020.v11.n2.a2","DOIUrl":"https://doi.org/10.4310/joc.2020.v11.n2.a2","url":null,"abstract":"Edelman and Greene constructed a correspondence between reduced words of the reverse permutation and standard Young tableaux. We prove that for any reduced word the shape of the region of the insertion tableau containing the smallest possible entries evolves exactly as the upper-left component of the permutation's (Rothe) diagram. Properties of the Edelman-Greene bijection restricted to 132-avoiding and 2143-avoiding permutations are presented. We also consider the Edelman-Greene bijection applied to non-reduced words.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"113 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80204435","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}
Pub Date : 2018-04-19DOI: 10.4310/JOC.2019.V10.N3.A2
Dun Qiu, J. Remmel
An ordered set partition of ${1,2,ldots,n}$ is a partition with an ordering on the parts. let $OP_{n,k}$ be the set of ordered set partitions of $[n]$ with $k$ blocks, Godbole, Goyt, Herdan and Pudwell defined $OP_{n,k}(sigma)$ to be the set of ordered set partitions in $OP_{n,k}$ avoiding a permutation pattern $sigma$ and obtained the formula for $|OP_{n,k}(sigma)|$ when the pattern $sigma$ is of length $2$. Later, Chen, Dai and Zhou found a formula algebraically for $|OP_{n,k}(sigma)|$ when the pattern $sigma$ is of length $3$. In this paper, we define a new pattern avoidance for the set $OP_{n,k}$, called $WOP_{n,k}(sigma)$, which includes the questions proposed by Godbole textit{et al.} We obtain formulas for $|WOP_{n,k}(sigma)|$ combinatorially for any $sigma$ of length $leq 3$. We also define 3 kinds of descent statistics on ordered set partitions and study the distribution of the descent statistics on $WOP_{n,k}(sigma)$ for $sigma$ of length $leq 3$.
{"title":"Patterns in words of ordered set partitions","authors":"Dun Qiu, J. Remmel","doi":"10.4310/JOC.2019.V10.N3.A2","DOIUrl":"https://doi.org/10.4310/JOC.2019.V10.N3.A2","url":null,"abstract":"An ordered set partition of ${1,2,ldots,n}$ is a partition with an ordering on the parts. let $OP_{n,k}$ be the set of ordered set partitions of $[n]$ with $k$ blocks, Godbole, Goyt, Herdan and Pudwell defined $OP_{n,k}(sigma)$ to be the set of ordered set partitions in $OP_{n,k}$ avoiding a permutation pattern $sigma$ and obtained the formula for $|OP_{n,k}(sigma)|$ when the pattern $sigma$ is of length $2$. Later, Chen, Dai and Zhou found a formula algebraically for $|OP_{n,k}(sigma)|$ when the pattern $sigma$ is of length $3$. \u0000In this paper, we define a new pattern avoidance for the set $OP_{n,k}$, called $WOP_{n,k}(sigma)$, which includes the questions proposed by Godbole textit{et al.} We obtain formulas for $|WOP_{n,k}(sigma)|$ combinatorially for any $sigma$ of length $leq 3$. We also define 3 kinds of descent statistics on ordered set partitions and study the distribution of the descent statistics on $WOP_{n,k}(sigma)$ for $sigma$ of length $leq 3$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"18 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80471256","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}
Pub Date : 2018-04-11DOI: 10.4310/joc.2020.v11.n1.a2
Louis DeBiasio, Andr'as Gy'arf'as, Robert A. Krueger, Mikl'os Ruszink'o, G'abor N. S'arkozy
It is well-known that in every $r$-coloring of the edges of the complete bipartite graph $K_{n,n}$ there is a monochromatic connected component with at least ${2nover r}$ vertices. It would be interesting to know whether we can additionally require that this large component be balanced; that is, is it true that in every $r$-coloring of $K_{n,n}$ there is a monochromatic component that meets both sides in at least $n/r$ vertices? Over forty years ago, Gy'arf'as and Lehel and independently Faudree and Schelp proved that any $2$-colored $K_{n,n}$ contains a monochromatic $P_n$. Very recently, Buci'c, Letzter and Sudakov proved that every $3$-colored $K_{n,n}$ contains a monochromatic connected matching (a matching whose edges are in the same connected component) of size $lceil n/3 rceil$. So the answer is strongly "yes" for $1leq rleq 3$. We provide a short proof of (a non-symmetric version of) the original question for $1leq rleq 3$; that is, every $r$-coloring of $K_{m,n}$ has a monochromatic component that meets each side in a $1/r$ proportion of its part size. Then, somewhat surprisingly, we show that the answer to the question is "no" for all $rge 4$. For instance, there are $4$-colorings of $K_{n,n}$ where the largest balanced monochromatic component has $n/5$ vertices in both partite classes (instead of $n/4$). Our constructions are based on lower bounds for the $r$-color bipartite Ramsey number of $P_4$, denoted $f(r)$, which is the smallest integer $ell$ such that in every $r$-coloring of the edges of $K_{ell,ell}$ there is a monochromatic path on four vertices. Furthermore, combined with earlier results, we determine $f(r)$ for every value of $r$.
{"title":"Monochromatic balanced components, matchings, and paths in multicolored complete bipartite graphs","authors":"Louis DeBiasio, Andr'as Gy'arf'as, Robert A. Krueger, Mikl'os Ruszink'o, G'abor N. S'arkozy","doi":"10.4310/joc.2020.v11.n1.a2","DOIUrl":"https://doi.org/10.4310/joc.2020.v11.n1.a2","url":null,"abstract":"It is well-known that in every $r$-coloring of the edges of the complete bipartite graph $K_{n,n}$ there is a monochromatic connected component with at least ${2nover r}$ vertices. It would be interesting to know whether we can additionally require that this large component be balanced; that is, is it true that in every $r$-coloring of $K_{n,n}$ there is a monochromatic component that meets both sides in at least $n/r$ vertices? \u0000Over forty years ago, Gy'arf'as and Lehel and independently Faudree and Schelp proved that any $2$-colored $K_{n,n}$ contains a monochromatic $P_n$. Very recently, Buci'c, Letzter and Sudakov proved that every $3$-colored $K_{n,n}$ contains a monochromatic connected matching (a matching whose edges are in the same connected component) of size $lceil n/3 rceil$. So the answer is strongly \"yes\" for $1leq rleq 3$. \u0000We provide a short proof of (a non-symmetric version of) the original question for $1leq rleq 3$; that is, every $r$-coloring of $K_{m,n}$ has a monochromatic component that meets each side in a $1/r$ proportion of its part size. Then, somewhat surprisingly, we show that the answer to the question is \"no\" for all $rge 4$. For instance, there are $4$-colorings of $K_{n,n}$ where the largest balanced monochromatic component has $n/5$ vertices in both partite classes (instead of $n/4$). Our constructions are based on lower bounds for the $r$-color bipartite Ramsey number of $P_4$, denoted $f(r)$, which is the smallest integer $ell$ such that in every $r$-coloring of the edges of $K_{ell,ell}$ there is a monochromatic path on four vertices. Furthermore, combined with earlier results, we determine $f(r)$ for every value of $r$.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"66 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89743893","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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