arXiv subjects
Publications and source records attributed to András Gyárfás.
We study how many comparability subgraphs are needed to partition the edge set of a perfect graph. We show that many classes of perfect graphs can be partitioned into (at most) two comparability subgraphs and this holds for almost all perfect graphs. On the other hand, we prove that for interval graphs an arbitrarily large number of comparability subgraphs might be necessary.
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}.
We call a proper edge coloring of a graph $G$ a B-coloring if every 4-cycle of $G$ is colored with four different colors. Let $q_B(G)$ denote the smallest number of colors needed for a B-coloring of $G$. Motivated by earlier papers on B-colorings, here we consider $q_B(G)$ for planar and outerplanar graphs in terms of the maximum degree $Δ= Δ(G)$. We prove that $q_B(G)\le 2Δ+8$ for planar graphs, $q_B(G)\le 2Δ$ for bipartite planar graphs and $q_B(G)\le Δ+1$ for outerplanar graphs with $Δ\ge 4$. We conjecture that, for $Δ$ sufficiently large, $q_B(G)\le 2Δ(G)$ for planar $G$ and $q_B(G)\le Δ(G)$ for outerplanar $G$.
Assume that $R_1,R_2,\dots,R_t$ are disjoint parallel lines in the plane. A $t$-interval (or $t$-track interval) is a set that can be written as the union of $t$ closed intervals, each on a different line. It is known that pairwise intersecting $2$-intervals can be pierced by two points, one from each line. However, it is not true that every set of pairwise intersecting $3$-intervals can be pierced by three points, one from each line. For $k\ge 3$, Kaiser and Rabinovich asked whether $k$-wise intersecting $t$-intervals can be pierced by $t$ points, one from each line. Our main result provides an asymptotic answer: in any set $S_1,\dots,S_n$ of $k$-wise intersecting $t$-intervals, at least $\frac{k-1}{k+1}n$ can be pierced by $t$ points, one from each line. We prove this in a more general form, replacing intervals by subtrees of a tree. This leads to questions and results on covering vertices of edge-colored complete graphs by vertices of monochromatic cliques having distinct colors, where the colorings are chordal, or more generally induced $C_4$-free graphs. For instance, we show that if the edges of a complete graph $K_n$ are colored with red or blue so that both color classes are induced $C_4$-free, then at least ${4n\over 5}$ vertices can be covered by a red and a blue clique, and this is best possible. We conclude by pointing to new Ramsey-type problems emerging from these restricted colorings.
Two independent edges in ordered graphs can be nested, crossing or separated. These relations define six types of subgraphs, depending on which relations are forbidden. We refine a remark by Erdős and Rado that every 2-coloring of the edges of a complete graph contains a monochromatic spanning tree. We show that forbidding one relation we always have a monochromatic (non-nested, non-crossing, non-separated) spanning tree in a 2-edge-colored ordered complete graph. On the other hand, if two relations are forbidden, then it is possible that we have monochromatic (nested, separated, crossing) subtrees of size only half the number of vertices. The existence of a monochromatic non-nested spanning tree in 2-colorings of ordered complete graphs verifies a more general conjecture for twisted drawings. Our second subject is to refine the Ramsey number of matchings for ordered complete graphs. Cockayne and Lorimer proved that for given positive integers t, n, m=(t-1)(n-1)+2n is the smallest integer with the following property: every t-coloring of the edges of a complete graph Km contains a monochromatic matching with n edges. We conjecture a strengthening: t-colored ordered complete graphs on m vertices contain monochromatic non-nested and also non-separated matchings with n edges. We prove this conjecture for some special cases. (i) Every t-colored ordered complete graph on t+3 vertices contains a monochromatic non-nested matching of size two. (ii) Every 2-colored ordered complete graph on 3n-1 vertices contains a monochromatic non-separated matching of size n. For nested, separated, and crossing matchings the situation is different. The smallest m ensuring a monochromatic matching of size n in every t-coloring is 2(t(n-1))+1) in the first two cases and one less in the third case.
The notion of cross intersecting set pair system of size $m$, $\Big(\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m\Big)$ with $A_i\cap B_i=\emptyset$ and $A_i\cap B_j\ne\emptyset$, was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that $m\le {a+b\choose a}$ if $|A_i|\le a$ and $|B_i|\le b$ for each $i$. Our central problem is to see how this bound changes with the additional condition $|A_i\cap B_j|=1$ for $i\ne j$. Such a system is called $1$-cross intersecting. We show that the maximum size of a $1$-cross intersecting set pair system is -- at least $5^{n/2}$ for $n$ even, $a=b=n$, -- equal to $\bigl(\lfloor\frac{n}{2}\rfloor+1\bigr)\bigl(\lceil\frac{n}{2}\rceil+1\bigr)$ if $a=2$ and $b=n\ge 4$, -- at most $|\cup_{i=1}^m A_i|$, -- asymptotically $n^2$ if $\{A_i\}$ is a linear hypergraph ($|A_i\cap A_j|\le 1$ for $i\ne j$), -- asymptotically ${1\over 2}n^2$ if $\{A_i\}$ and $\{B_i\}$ are both linear hypergraphs.
A {\em special four-cycle } $F$ in a triple system consists of four triples {\em inducing } a $C_4$. This means that $F$ has four special vertices $v_1,v_2,v_3,v_4$ and four triples in the form $w_iv_iv_{i+1}$ (indices are understood $\pmod 4$) where the $w_j$s are not necessarily distinct but disjoint from $\{v_1,v_2,v_3,v_4\}$. There are seven non-isomorphic special four-cycles, their family is denoted by $\cal{F}$. Our main result implies that the Turán number $\text{ex}(n,{\cal{F}})=Θ(n^{3/2})$. In fact, we prove more, $\text{ex}(n,\{F_1,F_2,F_3\})=Θ(n^{3/2})$, where the $F_i$-s are specific members of $\cal{F}$. This extends previous bounds for the Turán number of triple systems containing no Berge four cycles. We also study $\text{ex}(n,{\cal{A}})$ for all ${\cal{A}}\subseteq {\cal{F}}$. For 16 choices of $\cal{A}$ we show that $\text{ex}(n,{\cal{A}})=Θ(n^{3/2})$, for 92 choices of $\cal{A}$ we find that $\text{ex}(n,{\cal{A}})=Θ(n^2)$ and the other 18 cases remain unsolved.
A \textit{linear $3$-graph}, $H = (V, E)$, is a set, $V$, of vertices together with a set, $E$, of $3$-element subsets of $V$, called edges, so that any two distinct edges intersect in at most one vertex. The linear Turán number, ${\rm ex}(n,F)$, is the maximum number of edges in a linear $3$-graph $H$ with $n$ vertices containing no copy of $F$. We focus here on the \textit{crown}, $C$, which consists of three pairwise disjoint edges (jewels) and a fourth edge (base) which intersects all of the jewels. Our main result is that every linear $3$-graph with minimum degree at least $4$ contains a crown. This is not true if $4$ is replaced by $3$. In fact the known bounds of the Turán number are \[ 6 \left\lfloor{\frac{n - 3}{4}}\right\rfloor \leq {\rm ex}(n, C) \leq 2n, \] and in the construction providing the lower bound all but three vertices have degree $3$. We conjecture that ${\rm ex}(n, C) \sim \frac{3n}{2}$ but even if this were known it would not imply our main result. Our second result is a step towards a possible proof of ${\rm ex}(n,C) \leq \frac{3n}{2}$ (i.e., determining it within a constant error). We show that a minimal counterexample to this statement must contain certain configurations with $9$ edges and we conjecture that all of them lead to contradiction.
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.
A path-matching of order $p$ is a vertex disjoint union of nontrivial paths spanning $p$ vertices. Burr and Roberts, and Faudree and Schelp determined the 2-color Ramsey number of path-matchings. In this paper we study the multicolor Ramsey number of path-matchings. Given positive integers $r, p_1, \dots, p_r$, define $R^{PM}(p_1, \dots, p_r)$ to be the smallest integer $n$ such that in any $r$-coloring of the edges of $K_n$ there exists a path-matching of color $i$ and order at least $p_i$ for some $i\in [r]$. Our main result is that for $r\geq 2$ and $p_1\geq \dots\geq p_r\geq 2$, if $p_1\geq 2r-2$, then \[R^{PM}(p_1, \dots, p_r)= p_1- (r-1) + \sum_{i=2}^{r}\left\lceil\frac{p_i}{3}\right\rceil.\] Perhaps surprisingly, we show that when $p_1<2r-2$, it is possible that $R^{PM}(p_1, \dots, p_r)$ is larger than $p_1- (r-1) + \sum_{i=2}^{r}\left\lceil\frac{p_i}{3}\right\rceil$, but in any case we determine the correct value to within a constant (depending on $r$); i.e. \[p_1- (r-1) + \sum_{i=2}^{r}\left\lceil\frac{p_i}{3}\right\rceil \leq R^{PM}(p_1, \dots, p_r)\leq \left\lceil p_1-\frac{r}{3}+\sum_{i=2}^r\frac{p_i}{3}\right\rceil.\] As a corollary we get that in every $r$-coloring of $K_n$ there is a monochromatic path-matching of order at least $3\left\lfloor\frac{n}{r+2}\right\rfloor$, which is essentially best possible. We also determine $R^{PM}(p_1, \dots, p_r)$ in all cases when the number of colors is at most 4. The proof of the main result uses a minimax theorem for path-matchings derived from a result of Las Vergnas (extending Tutte's 1-factor theorem) to show that the value of $R^{PM}(p_1, \dots, p_r)$ depends on the block sizes in covering designs (which can be also formulated in terms of monochromatic 1-cores in colored complete graphs). Then we obtain the result above by giving estimates on the block sizes in covering designs in the arbitrary (non-uniform) case.
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 ${2n\over r}$ 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, Gyárfás and Lehel and independently Faudree and Schelp proved that any $2$-colored $K_{n,n}$ contains a monochromatic $P_n$. Very recently, Bucić, 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 $1\leq r\leq 3$. We provide a short proof of (a non-symmetric version of) the original question for $1\leq r\leq 3$; 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 $r\ge 4$. 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 $r$.
A Gallai coloring is an edge coloring that avoids triangles colored with three different colors. Given integers $e_1\ge e_2 \ge \dots \ge e_k$ with $\sum_{i=1}^ke_i={n \choose 2}$ for some $n$, does there exist a Gallai $k$-coloring of $K_n$ with $e_i$ edges in color $i$? In this paper, we give several sufficient conditions and one necessary condition to guarantee a positive answer to the above question. In particular, we prove the existence of a Gallai-coloring if $e_1-e_k\le 1$ and $k \le \lfloor n/2\rfloor$. We prove that for any integer $k\ge 3$ there is a (unique) integer $g(k)$ with the following property: there exists a Gallai $k$-coloring of $K_n$ with $e_i$ edges in color $i$ for every $e_1\le\dots \le e_k$ satisfying $\sum_{i=1}^ke_i={n\choose 2}$, if and only if $n\ge g(k)$. We show that $g(3)=5$, $g(4)=8$, and $2k-2\le g(k)\le 8k^2+1$ for every $k\ge 3$.
We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle --- three different subsets $A,B,C\subseteq [n]$ such that $A\cap B$, $A\cap C$, and $B\cap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts. We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp (for fixed $k$) when $G=C_k, P_k, S_k$, a cycle, path, or star with $k$ edges. Additional bounds are given for $G=C_4$ and $G=S_3$.
In this paper we generalize the concept of uniquely $K_r$-saturated graphs to hypergraphs. Let $K_r^{(k)}$ denote the complete $k$-uniform hypergraph on $r$ vertices. For integers $k,r,n$ such that $2\le k r$. This is in contrast to the case $k=2$ and $r=3$ where only the Moore graphs of diameter two have this property. Our other construction keeps $n-r$ fixed; in this case we show that for any fixed $k\ge 2$ there can only be finitely many examples. We give a range for $n$ where these hypergraphs exist. For $n-r=1$ the range is completely determined: $k+1\le n \le {(k+2)^2\over 4}$. For larger values of $n-r$ the upper end of our range reaches approximately half of its upper bound. The lower end depends on the chromatic number of certain Johnson graphs.
A hypergraph is linear if any two edges intersect in at most one vertex. For a fixed $k$-uniform family ${\cal{F}}$ of hypergraphs, the linear Turán number ${\rm ex}_{\rm lin}(n,{\cal{F}})$ is the maximum number of edges in a $k$-uniform linear hypergraph $\mathcal H$ on $n$ vertices that does not contain any member of ${\cal{F}}$ as a subhypergraph. For $k\ge 2$ the $k$-fan $F^k$ is the $k$-uniform linear hypergraph having $k$ edges $f_1,\dots,f_k$ pairwise intersecting in the same vertex $v$ and an additional edge $g$ intersecting all $f_i$ in a vertex different from $v$. We prove the following extension of Mantel's theorem $${\rm ex}_{\rm lin}(n,F^k)\le {n^2 / k^2}.$$ Moreover, $|{\mathcal H}|=n^2/k^2$ holds if and only if $n\equiv 0\pmod k$ and $\mathcal H$ is a transversal design on $n$ points with $k$ groups. We also study ${\rm ex}_{\rm lin}(n,{\cal{F}})$ where $\cal{F}$ is any subset of the three linear triple systems with four triples on at most seven points.
A well-known special case of a conjecture attributed to Ryser states that k-partite intersecting hypergraphs have transversals of at most k-1 vertices. An equivalent form was formulated by Gyárfás: if the edges of a complete graph K are colored with k colors then the vertex set of K can be covered by at most k-1 sets, each connected in some color. It turned out that the analogue of the conjecture for hypergraphs can be answered: Z. Király proved that in every k-coloring of the edges of the r-uniform complete hypergraph K^r (r >= 3), the vertex set of K^r can be covered by at most $\lceil k/r \rceil$ sets, each connected in some color. Here we investigate the analogue problem for complete r-uniform r-partite hypergraphs. An edge coloring of a hypergraph is called spanning if every vertex is incident to edges of any color used in the coloring. We propose the following analogue of Ryser conjecture. In every spanning (r+t)-coloring of the edges of a complete r-uniform r-partite hypergraph, the vertex set can be covered by at most t+1 sets, each connected in some color. Our main result is that the conjecture is true for 1 <= t <= r-1. We also prove a slightly weaker result for t >= r, namely that t+2 sets, each connected in some color, are enough to cover the vertex set. To build a bridge between complete r-uniform and complete r-uniform r-partite hypergraphs, we introduce a new notion. A hypergraph is complete r-uniform (r,l)-partite if it has all r-sets that intersect each partite class in at most l vertices. Extending our results achieved for l=1, we prove that for any r >= 3, 2 <= l <= r, k >= 1+r-l, in every spanning k-coloring of the edges of a complete r-uniform (r,l)-partite hypergraph, the vertex set can be covered by at most 1+\lfloor \frac{k-r+\ell-1}{\ell}\rfloor sets, each connected in some color.
Suppose that $k$ is a non-negative integer and a bipartite multigraph $G$ is the union of $$N=\left\lfloor \frac{k+2}{k+1}n\right\rfloor -(k+1)$$ matchings $M_1,\dots,M_N$, each of size $n$. We show that $G$ has a rainbow matching of size $n-k$, i.e. a matching of size $n-k$ with all edges coming from different $M_i$'s. Several choices of parameters relate to known results and conjectures.
Suppose that $T$ is an acyclic $r$-uniform hypergraph, with $r\ge 2$. We define the ($t$-color) chromatic Ramsey number $χ(T,t)$ as the smallest $m$ with the following property: if the edges of any $m$-chromatic $r$-uniform hypergraph are colored with $t$ colors in any manner, there is a monochromatic copy of $T$. We observe that $χ(T,t)$ is well defined and $$\left\lceil {R^r(T,t)-1\over r-1}\right \rceil +1 \le χ(T,t)\le |E(T)|^t+1$$ where $R^r(T,t)$ is the $t$-color Ramsey number of $H$. We give linear upper bounds for $χ(T,t)$ when T is a matching or star, proving that for $r\ge 2, k\ge 1, t\ge 1$, $χ(M_k^r,t)\le (t-1)(k-1)+2k$ and $χ(S_k^r,t)\le t(k-1)+2$ where $M_k^r$ and $S_k^r$ are, respectively, the $r$-uniform matching and star with $k$ edges. The general bounds are improved for $3$-uniform hypergraphs. We prove that $χ(M_k^3,2)=2k$, extending a special case of Alon-Frankl-Lovász' theorem. We also prove that $χ(S_2^3,t)\le t+1$, which is sharp for $t=2,3$. This is a corollary of a more general result. We define $H^{[1]}$ as the 1-intersection graph of $H$, whose vertices represent hyperedges and whose edges represent intersections of hyperedges in exactly one vertex. We prove that $χ(H)\le χ(H^{[1]})$ for any $3$-uniform hypergraph $H$ (assuming $χ(H^{[1]})\ge 2$). The proof uses the list coloring version of Brooks' theorem.