arXiv subjects
Publications and source records attributed to Louis DeBiasio.
Related to a bounded-diameter bipartite analogue of the Henderson--Ryser conjecture, DeBiasio, Kamel, McCourt, and Sheats proved that the vertices of every $2$-colored complete bipartite graph can be covered by two monochromatic subgraphs, each of diameter at most four. We improve this bound on the diameter to the best possible value of {\em three}.
Treglown conjectured (in a complementary form) that for every positive integer $k$, every digraph $D$ satisfying $\min\{d^+(v),d^-(v)\}\le k-1$ for all $v\in V(D)$ has an equitable acyclic $k$-coloring. If true, this would imply the acyclic coloring versions of the Hajnal-Szemerédi theorem for digraphs proved by Czygrinow, DeBiasio, Kierstead, and Molla (which in turn imply the original Hajnal-Szemerédi theorem for graphs). As it turns out, there is a simple reduction implicit in Aboulker, Oijid, Petit, Rocton, and Simon which surprisingly shows that Treglown's conjecture (and thus the results of Czygrinow, DeBiasio, Kierstead, and Molla) follows directly from the original Hajnal--Szemerédi theorem for graphs. We slightly modify the reduction in order to show that there exists a polynomial time algorithm for finding an equitable acyclic $k$-coloring in such a digraph.
The wheel $W_{k}$ is the graph on $k+1$ vertices consisting of a vertex joined to a cycle of length $k$, and we say that $W_k$ is an even wheel if $k$ is even. Mao, Wang, Magnant, Schiermeyer proved that the Ramsey number of $W_{2n}$ is between $4n+1$ and $12n-2$. We improve both of these bounds, showing that $5n-\frac{1+(-1)^{n}}{2}\leq R(W_{2n})\leq 8n+664$ for all integers $n\geq 2$. The main focus of the paper concerns two general results on the Ramsey numbers of stars versus even wheels and even cycles versus even wheels, from which the above bounds are obtained as a corollary. That is, we asymptotically determine $R(K_{1,m}, W_{2n})$ and $R(C_{2m}, W_{2n})$ for all sufficiently large $m$ and $n$, both of which were open problems for most regimes. As for odd wheels, we note that the analogous values for stars versus odd wheels and odd cycles versus odd wheels were already known exactly, from which it follows that $6n+4=R(K_{1,2n+1}, W_{2n+1})\leq R(W_{2n+1})\leq 2\cdot R(C_{2n+1}, W_{2n+1})=12n+2$. Very recently, Zhang and Chen improved the upper bound to $R(W_{2n+1})\leq \frac{32n}{3}+O(1)$. We are able to refine their proof, further improving the upper bound to $R(W_{2n+1})\leq 10n+O(1)$.
The $r$-color size-Ramsey number of a graph $H$, denoted by $\widehat{R}_r(H)$, is the minimum number of edges in a graph $G$ having the property that every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H$. Krivelevich proved that $\widehat{R}_r(P_{m+1})=Ω(r^2m)$ where $P_{m+1}$ is the path on $m$ edges. He explains that his proof actually applies to any connected graph $H$ with $m$ edges and vertex cover number larger than $\sqrt{m}$. He also notes that some restriction on the vertex cover number is necessary since the star with $m$ edges, $K_{1,m}$, has vertex cover number 1 and satisfies $\widehat{R}_r(K_{1,m})=r(m-1)+1$. We prove that the star is actually the only exception; that is, $\widehat{R}_r(H)=Ω(r^2m)$ for every non-star connected graph $H$ with $m$ edges. We also prove a strengthening of this result for trees. It follows from results of Beck and Dellamonica that $\widehat{R}_2(T)=Θ(β(T))$ for every tree $T$ with bipartition $\{V_1, V_2\}$ and $β(T)=|V_1|\max\{d(v):v\in V_1\}+|V_2|\max\{d(v):v\in V_2\}$. We prove that $\widehat{R}_r(T)=Ω(r^2β(T))$ for every tree $T$, again with the exception of the star. Additionally, we prove that for the family of non-star trees $T$ with $β(T)=Ω(n_1n_2)$ (which includes all non-star trees of linear maximum degree and all trees of radius 2 for example) we have $\widehat{R}_r(T)=Θ(r^2β(T))$.
Let $F_n$ be the graph on $2n+1$ vertices consisting of $n$ triangles meeting at a single vertex. After a number of improvements over the years, it is currently known that the Ramsey number of $F_n$ is between $4.5n-5$ (Chen, Yu, Zhao) and $(5+\frac{1}{6})n+O(1)$ (Dvo{ř}{á}k and Metrebian). We improve both of these bounds as follows $$4.732n\approx (3+\sqrt{3})n-8< R(F_n)\leq (5+o(1))n.$$ Additionally, as it relates to the lower bound on $R(F_n)$ (and for which nothing was known when $n< m< n(n-1)$), we determine the Ramsey numbers of stars vs.~fans, within a constant, as follows $$R(K_{1,m}, F_{n})= \begin{cases} m+2n-\frac{1+(-1)^{m}}{2}, & m\leq n \frac{3m+\sqrt{m^2+8n^2}}{2}+Θ(1), & m>n \end{cases}. $$ In particular, we have $R(K_{1,2n}, F_n)=(3+\sqrt{3})n+Θ(1)$.
The Pósa--Seymour conjecture determines the minimum degree threshold for forcing the $k$th power of a Hamilton cycle in a graph. After numerous partial results, Komlós, Sárközy and Szemerédi proved the conjecture for sufficiently large graphs. In this paper we focus on the analogous problem for digraphs and for oriented graphs. We asymptotically determine the minimum total degree threshold for forcing the square of a Hamilton cycle in a digraph. We also give a conjecture on the corresponding threshold for $k$th powers of a Hamilton cycle more generally. For oriented graphs, we provide a minimum semi-degree condition that forces the $k$th power of a Hamilton cycle; although this minimum semi-degree condition is not tight, it does provide the correct order of magnitude of the threshold. Turán-type problems for oriented graphs are also discussed.
In 1960, Ghouila-Houri proved that every strongly connected directed graph $G$ on $n$ vertices with minimum degree at least $n$ contains a directed Hamilton cycle. We asymptotically generalize this result by proving the following: every directed graph $G$ on $n$ vertices and with minimum degree at least $(1+o(1))n$ contains every orientation of a Hamilton cycle, except for the directed Hamilton cycle in the case when $G$ is not strongly connected. In fact, this minimum degree condition forces every orientation of a cycle in $G$ of every possible length, other than perhaps the directed cycles.
In 1967, Gerencsér and Gyárfás proved a result which is considered the starting point of graph-Ramsey theory: In every 2-coloring of $K_n$ there is a monochromatic path on $\lceil(2n+1)/3\rceil$ vertices, and this is best possible. There have since been hundreds of papers on graph-Ramsey theory with some of the most important results being motivated by a series of conjectures of Burr and Erd\H os regarding the Ramsey numbers of trees, graphs with bounded maximum degree, and graphs with bounded degeneracy. In 1993, Erd\H os and Galvin \cite{EG} began the investigation of a countably infinite analogue of the Gerencsér and Gyárfás result: What is the largest $d$ such that in every $2$-coloring of $K_\mathbb{N}$ there is a monochromatic infinite path with upper density at least $d$. Erd\H os and Galvin showed that $2/3\leq d\leq 8/9$, and after a series of recent improvements, it was finally shown that $d={(12+\sqrt{8})}/{17}$. This paper begins a systematic study of quantitative countably infinite graph-Ramsey theory, focusing on infinite analogues of the Burr-Erdős conjectures. We obtain some results which are analogous to what is known in finite case, and other (unexpected) results which have no analogue in the finite case.
K\H onig's theorem says that the vertex cover number of every bipartite graph is at most its matching number (in fact they are equal since, trivially, the matching number is at most the vertex cover number). An equivalent formulation of K\H onig's theorem is that in every $2$-colouring of the edges of a graph $G$, the number of monochromatic components needed to cover the vertex set of $G$ is at most the independence number of $G$. We prove the following strengthening of K\H onig's theorem: In every $2$-colouring of the edges of a graph $G$, the number of monochromatic subgraphs of bounded diameter needed to cover the vertex set of $G$ is at most the independence number of $G$.
We prove a strong dichotomy result for countably-infinite oriented graphs; that is, we prove that for all countably-infinite oriented graphs $G$, either (i) there is a countably-infinite tournament $K$ such that $G\not\subseteq K$, or (ii) every countably-infinite tournament contains a \emph{spanning} copy of $G$. Furthermore, we are able to give a concise characterization of such oriented graphs. Our characterization becomes even simpler in the case of transitive acyclic oriented graphs (i.e. strict partial orders). For uncountable oriented graphs, we are able to extend the dichotomy result mentioned above to all regular cardinals $κ$; however, we are only able to provide a concise characterization in the case when $κ=\aleph_1$.
The $r$-color size-Ramsey number of a $k$-uniform hypergraph $H$, denoted by $\hat{R}_r(H)$, is the minimum number of edges in a $k$-uniform hypergraph $G$ such that for every $r$-coloring of the edges of $G$ there exists a monochromatic copy of $H$. In the case of $2$-uniform paths $P_n$, it is known that $Ω(r^2n)=\hat{R}_r(P_n)=O((r^2\log r)n)$ with the best bounds essentially due to Krivelevich. In a recent breakthrough result, Letzter, Pokrovskiy, and Yepremyan gave a linear upper bound on the $r$-color size-Ramsey number of the $k$-uniform tight path $P_{n}^{(k)}$; i.e. $\hat{R}_r(P_{n}^{(k)})=O_{r,k}(n)$. Winter gave the first non-trivial lower bounds on the 2-color size-Ramsey number of $P_{n}^{(k)}$ for $k\geq 3$; i.e. $\hat{R}_2(P_{n}^{(3)})\geq \frac{8}{3}n-O(1)$ and $\hat{R}_2(P_{n}^{(k)})\geq \lceil\log_2(k+1)\rceil n-O_k(1)$ for $k\geq 4$. We consider the problem of giving a lower bound on the $r$-color size-Ramsey number of $P_{n}^{(k)}$ (for fixed $k$ and growing $r$). Our main result is that $\hat{R}_r(P_n^{(k)})=Ω_k(r^kn)$ which generalizes the best known lower bound for graphs mentioned above. One of the key elements of our proof is a determination of the correct order of magnitude of the $r$-color size-Ramsey number of every sufficiently short tight path; i.e. $\hat{R}_r(P_{k+m}^{(k)})=Θ_k(r^m)$ for all $1\leq m\leq k$. All of our results generalize to $\ell$-overlapping $k$-uniform paths $P_{n}^{(k, \ell)}$. In particular we note that when $1\leq \ell\leq \frac{k}{2}$, we have $Ω_k(r^{2}n)=\hat{R}_r(P_{n}^{(k, \ell)})=O((r^2\log r)n)$ which essentially matches the best known bounds for graphs mentioned above. Additionally, in the case $k=3$, $\ell=2$, and $r=2$, we give a more precise estimate which implies $\hat{R}_2(P^{(3)}_{n})\geq \frac{28}{9}n-O(1)$, improving on the above-mentioned lower bound of Winter in the case $k=3$.
The balanced double star on $2n+2$ vertices, denoted $S_{n,n}$, is the tree obtained by joining the centers of two disjoint stars each having $n$ leaves. Let $R_r(G)$ be the smallest integer $N$ such that in every $r$-coloring of the edges of $K_N$ there is a monochromatic copy of $G$, and let $R_r^{\mathrm{bip}}(G)$ be the smallest integer $N$ such that in every $r$-coloring of the edges of $K_{N,N}$ there is a monochromatic copy of $G$. It is known that $R_2(S_{n,n})=3n+2$ and $R_2^{\mathrm{bip}}(S_{n,n})=2n+1$ \cite{HJ}, but very little is known about $R_r(S_{n,n})$ and $R^{\mathrm{bip}}_r(S_{n,n})$ when $r\geq 3$ (other than the bounds which follow from considerations on the number of edges in the majority color class). In this paper we prove the following for all $n\geq 1$ (where the lower bounds are adapted from existing examples): \[(r-1)2n+1\leq R_r(S_{n,n})\leq (r-\frac{1}{2})(2n+2)-1,\]and \[(2r-4)n+1\leq R^{\mathrm{bip}}_r(S_{n,n})\leq (2r-3+\frac{2}{r}+O(\frac{1}{r^2}))n.\] These bounds are similar to the best known bounds on $R_r(P_{2n+2})$ and $R_r^{\mathrm{bip}}(P_{2n+2})$, where $P_{2n+2}$ is a path on $2n+2$ vertices (which is also a balanced tree). We also give an example which improves the lower bound on $R^{\mathrm{bip}}_r(S_{n,n})$ when $r=3$ and $r=5$.
A result of Gyárfás says that for every $3$-coloring of the edges of the complete graph $K_n$, there is a monochromatic component of order at least $\frac{n}{2}$, and this is best possible when $4$ divides $n$. Furthermore, for all $k\geq 3$ and every $(k+1)$-coloring of the edges of the complete $k$-uniform hypergraph $K_n^{k}$, there is a monochromatic component of order at least $\frac{kn}{k+1}$ and this is best possible for all $n$. Recently, Guggiari and Scott and independently Rahimi proved a strengthening of the graph case in the result above which says that the same conclusion holds if $K_n$ is replaced by any graph on $n$ vertices with minimum degree at least $\frac{5n}{6}-1$; furthermore, this bound on the minimum degree is best possible. We prove a strengthening of the $k\geq 3$ case in the result above which says that the same conclusion holds if $K_n^k$ is replaced by any $k$-uniform hypergraph on $n$ vertices with minimum $(k-1)$-degree at least $\frac{kn}{k+1}-(k-1)$; furthermore, this bound on the $(k-1)$-degree is best possible.
A result of Gyárfás exactly determines the size of a largest monochromatic component in an arbitrary $r$-coloring of the complete $k$-uniform hypergraph $K_n^k$ when $k\geq 2$ and $r-1\leq k\leq r$. We prove a result which says that if one replaces $K_n^k$ in Gyárfás' theorem by any ``expansive'' $k$-uniform hypergraph on $n$ vertices (that is, a $k$-uniform hypergraph $H$ on $n$ vertices in which in which $e(V_1, \dots, V_k)>0$ for all disjoint sets $V_1, \dots, V_k\subseteq V(H)$ with $|V_i|>α$ for all $i\in [k]$), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on $r$ and $α$). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms. Gyárfás' result is equivalent to the dual problem of determining the smallest maximum degree of an arbitrary $r$-partite $r$-uniform hypergraph with $n$ edges in which every set of $k$ edges has a common intersection. In this language, our result says that if one replaces the condition that every set of $k$ edges has a common intersection with the condition that for every collection of $k$ disjoint sets $E_1, \dots, E_k\subseteq E(H)$ with $|E_i|>α$ for all $i\in [k]$ there exists $e_i\in E_i$ for all $i\in [k]$ such that $e_1\cap \dots \cap e_k\neq \emptyset$, then the maximum degree of $H$ is essentially the same (within a small error term depending on $r$ and $α$). We prove our results in this dual setting.
We prove that if $G$ is a $k$-partite graph on $n$ vertices in which all of the parts have order at most $n/r$ and every vertex is adjacent to at least a $1-1/r+o(1)$ proportion of the vertices in every other part, then $G$ contains the $(r-1)$-st power of a Hamiltonian cycle
We prove that for all graphs with at most $(3.75-o(1))n$ edges there exists a 2-coloring of the edges such that every monochromatic path has order less than $n$. This was previously known to be true for graphs with at most $2.5n-7.5$ edges. We also improve on the best-known lower bounds in the $r$-color case.
Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $ν(H)$, the vertex cover number is at most $(r-1)ν(H)$. This far reaching generalization of König's theorem is only known to be true for $r\leq 3$, or $ν(G)=1$ and $r\leq 5$. An equivalent formulation of Ryser's conjecture is that in every $r$-edge coloring of a graph $G$ with independence number $α(G)$, there exists at most $(r-1)α(G)$ monochromatic connected subgraphs which cover the vertex set of $G$. We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs. Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results.
A digraph is {\em $d$-dominating} if every set of at most $d$ vertices has a common out-neighbor. For all integers $d\geq 2$, let $f(d)$ be the smallest integer such that the vertices of every 2-edge-colored (finite or infinite) complete digraph (including loops) can be covered by the vertices of at most $f(d)$ monochromatic $d$-dominating subgraphs. Note that the existence of $f(d)$ is not obvious -- indeed, the question which motivated this paper was simply to determine whether $f(d)$ is bounded, even for $d=2$. We answer this question affirmatively for all $d\geq 2$, proving $4\leq f(2)\le 8$ and $2d\leq f(d)\le 2d\left(\frac{d^{d}-1}{d-1}\right)$ for all $d\ge 3$. We also give an example to show that there is no analogous bound for more than two colors. Our result provides a positive answer to a question regarding an infinite analogue of the Burr-Erdős conjecture on the Ramsey numbers of $d$-degenerate graphs. Moreover, a special case of our result is related to properties of $d$-paradoxical tournaments.