As a natural extension of the Four Color Theorem, Hajós conjectured that graphs containing no $K_5$-subdivision are 4-colorable. Any possible counterexample to this conjecture with minimum number of vertices is called a {\it Hajós graph}. Previous results show that Hajós graphs are 4-connected but not 5-connected. A $k$-separation in a graph $G$ is a pair $(G_1,G_2)$ of edge-disjoint subgraphs of $G$ such that $|V(G_1\cap G_2)|=k$, $G=G_1\cup G_2$, and $G_i\not\subseteq G_{3-i}$ for $i=1,2$. In this paper, we show that Hajós graphs do not admit a 4-separation $(G_1,G_2)$ such that $|V(G_1)|\ge 6$ and $G_1$ can be drawn in the plane with no edge crossings and all vertices in $V(G_1\cap G_2)$ incident with a common face. This is a step in our attempt to reduce Hajós' conjecture to the Four Color Theorem.
It was conjectured by Hajós that graphs containing no $K_5$-subdivision are 4-colorable. Previous results show that any possible minimum counterexample to Hajós' conjecture, called Hajós graph, is 4-connected but not 5-connected. In this paper, we show that if a Hajós graph admits a 4-cut or 5-cut with a planar side then the planar side must be small or contains a special wheel. This is a step in our effort to reduce Hajós' conjecture to the Four Color Theorem.
Hajós' conjecture states that an Eulerian graph of order n can be decomposed into at most (n-1)/2 edge-disjoint cycles. We describe preprocessing steps, heuristics and integer programming techniques that enable us to verify Hajós' conjecture for all Eulerian graphs with up to twelve nodes.
The paper designs five graph operations, and proves that every signed graph with chromatic number $q$ can be obtained from all-positive complete graphs $(K_q,+)$ by repeatedly applying these operations. This result gives a signed version of the Hajós theorem, emphasizing the role of all-positive complete graphs played in the class of signed graphs, as played in the class of unsigned graphs.
Hajnal and Szemeredi proved that every graph G with |G|=ks and minimum degree at least k(s-1) contains k vertex disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graph D with |D|=ks and minimum (total) degree at least 2k(s-1)-1 contains k vertex disjoint transitive tournaments on s vertices. Our result implies the Hajnal-Szemeredi Theorem, and the degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjec…
Let $G=(V,E)$ be a simple graph. A set $D\subseteq V$ is a strong dominating set of $G$, if for every vertex $x\in V\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number $γ_{st}(G)$ is defined as the minimum cardinality of a strong dominating set. In this paper, we study the strong domination number of Hajós sum and vertex-sum of two graphs.
We say that a graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. The seminal Hajnal--Szemerédi theorem characterises the minimum degree that ensures a graph $G$ contains a perfect $K_r$-packing. Balogh, Kostochka and Treglown proposed a degree sequence version of the Hajnal--Szemerédi theorem which, if true, gives a strengthening of the Hajnal--Szemerédi theorem. In this paper we prove this conjecture asymptotically. Another fundamental result in the area is the Alon--Yuster theorem which gives a minimum degree condit…
The Erdös-Hajnal conjecture states that for every graph $H$, there exists a constant $δ(H) > 0$ such that every graph $G$ with no induced subgraph isomorphic to $H$ has either a clique or a stable set of size at least $|V(G)|^{δ(H)}$. This paper is a survey of some of the known results on this conjecture.
In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that if $p(n) \gg (\log n / n)^{1/2}$, then asymptotically almost surely every subgraph of $G(n,p)$ with minimum degree at least $(2/3 + o(1))np$ contains a triangle packing that covers all but at most $O(p^{-2})$ vertices. Moreover, the assumption on $p$ is optimal up to the $(\log n)^{1/2}$ factor and the presence of the set of $O(p^{-2})$ uncovered vertices is indispensable. The main ingredient in the proof, which might be of independent interest, is an embedding theorem which…
A perfect $K_t$-matching in a graph $G$ is a spanning subgraph consisting of vertex disjoint copies of $K_t$. A classic theorem of Hajnal and Szemerédi states that if $G$ is a graph of order $n$ with minimum degree $δ(G) \ge (t-1)n/t$ and $t| n$, then $G$ contains a perfect $K_t$-matching. Let $G$ be a $t$-partite graph with vertex classes $V_1$,..., $V_t$ each of size $n$. We show that if every vertex $x \in V_i$ is joined to at least $((t-1)/t + γ)n $ vertices of $V_j$ for $i \ne j$, then $G$ contains a perfect $K_t$-matching, thus verifying a conjecture of Fisher asymptotically. Furthermore…
The celebrated Hajnal-Szemerédi theorem gives the precise minimum degree threshold that forces a graph to contain a perfect K_k-packing. Fischer's conjecture states that the analogous result holds for all multipartite graphs except for those formed by a single construction. Recently, we deduced an approximate version of this conjecture from new results on perfect matchings in hypergraphs. In this paper, we apply a stability analysis to the extremal cases of this argument, thus showing that the exact conjecture holds for any sufficiently large graph.
A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of the clique $K_r$ in $G$ covering every vertex of $G$. The famous Hajnal--Szemerédi theorem determines the minimum degree threshold for forcing a perfect $K_r$-tiling in a graph $G$. The notion of discrepancy appears in many branches of mathematics. In the graph setting, one assigns the edges of a graph $G$ labels from $\{-1,1\}$, and one seeks substructures $F$ of $G$ that have `high' discrepancy (i.e. the sum of the labels of the edges in $F$ is far from $0$). In this paper we determine the minimum degree thres…
We prove that for every graph $H$, there exists $\varepsilon>0$ such that every $n$-vertex graph with no vertex-minors isomorphic to $H$ has a pair of disjoint sets $A$, $B$ of vertices such that $|A|, |B|\ge \varepsilon n$ and $A$ is complete or anticomplete to $B$. We deduce this from recent work of Chudnovsky, Scott, Seymour, and Spirkl (2018). This proves the analog of the Erdős-Hajnal conjecture for vertex-minors.
Let $G$ be a tripartite graph with $N$ vertices in each vertex class. If each vertex is adjacent to at least $(2/3)N$ vertices in each of the other classes, then either $G$ contains a subgraph that consists of $N$ vertex-disjoint triangles or $G$ is a specific graph in which each vertex is adjacent to exactly $(2/3)N$ vertices in each of the other classes.
Any graph $G$ with chromatic number $k$ can be constructed by iteratively performing certain graph operations on a sequence of graphs starting with $K_k$, resulting in a variety of Hajós-type constructions for $G$. Finding such constructions for a given graph or family of graphs is a challenging task. We show that the basic steps in these Hajós-type constructions frequently result in the presence of an $S^1$-wedge summand in the neighborhood complex of the resulting graph. Our results imply that for a graph $G$ with a highly-connected neighborhood complex, the end behavior of the construction …
For a nondegenerate $r$-graph $F$, large $n$, and $t$ in the regime $[0, c_{F} n]$, where $c_F>0$ is a constant depending only on $F$, we present a general approach for determining the maximum number of edges in an $n$-vertex $r$-graph that does not contain $t+1$ vertex-disjoint copies of $F$. In fact, our method results in a rainbow version of the above result and includes a characterization of the extremal constructions. Our approach applies to many well-studied hypergraphs (including graphs) such as the edge-critical graphs, the Fano plane, the generalized triangles, hypergraph expansions, …
The Erdős--Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph $H$, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of $r$-uniform hypergraphs. A theorem of Rödl asserts that if an $n$-vertex graph is non-universal then it contains an almost homogeneous set (i.e one with edge density either very close to $0$ or $1$) of size $Ω(n)$. We prove that if a $3$-uniform hypergraph is non-universal then it contains an almost homog…
In 1963, Corrádi and Hajnal proved that for all $k\geq1$ and $n\geq3k$, every graph $G$ on $n$ vertices with minimum degree $δ(G)\geq2k$ contains $k$ disjoint cycles. The bound $δ(G) \geq 2k$ is sharp. Here we characterize those graphs with $δ(G)\geq2k-1$ that contain $k$ disjoint cycles. This answers the simple-graph case of Dirac's 1963 question on the characterization of $(2k-1)$-connected graphs with no $k$ disjoint cycles. Enomoto and Wang refined the Corrádi-Hajnal Theorem, proving the following Ore-type version: For all $k\geq1$ and $n\geq3k$, every graph $G$ on $n$ vertices contains $k…
The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every $n$-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least $e^{(\log n)^{1 - o(1)}}$. The dependence on the VC-dimension is hidden in the $o(1)$ term. This improves the general lower bound, $e^{c\sqrt{\log n}}$, due to Erdos and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial hereditary property. Our result is almost optimal and nearly matches…
We prove that every 3-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a set of order Omega(n^{1/3}log^2 n) which uses at most two colors, and this bound is tight up to a constant factor. This verifies a conjecture of Hajnal which is a case of the multicolor generalization of the well-known Erdős-Hajnal conjecture. We further establish a generalization of this result. For fixed positive integers s and r with s at most r, we determine a constant c_{r,s} such that the following holds. Every r-coloring of the edges of the complete graph on n vertices w…
For a graph $G$, let $χ(G)$ denote its chromatic number and $σ(G)$ denote the order of the largest clique subdivision in $G$. Let H(n) be the maximum of $χ(G)/σ(G)$ over all $n$-vertex graphs $G$. A famous conjecture of Hajós from 1961 states that $σ(G) \geq χ(G)$ for every graph $G$. That is, $H(n) \leq 1$ for all positive integers $n$. This conjecture was disproved by Catlin in 1979. Erdős and Fajtlowicz further showed by considering a random graph that $H(n) \geq cn^{1/2}/\log n$ for some absolute constant $c>0$. In 1981 they conjectured that this bound is tight up to a constant factor in t…
For every positive integer $k$, we show that every graph of order $n$ at least $3k$ with more than $$\max\{{2k-1\choose 2}+(2k-1)(n-(2k-1)),{3k-1\choose 2}+(n-(3k-1))\}$$ edges has $k$ vertex disjoint cycles, which is a best possible density version of a theorem of Corrádi and Hajnal.
A path (resp. cycle) decomposition of a graph $G$ is a set of edge-disjoint paths (resp. cycles) of $G$ that covers the edge set of $G$. Gallai (1966) conjectured that every graph on $n$ vertices admits a path decomposition of size at most $\lfloor (n+1)/2\rfloor$, and Hajós (1968) conjectured that every Eulerian graph on $n$ vertices admits a cycle decomposition of size at most $\lfloor (n-1)/2\rfloor$. Gallai's Conjecture was verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex of even degree, and Pyber (1996) verified …
For a positive integer r>=2, a K_r-factor of a graph is a collection vertex-disjoint copies of K_r which covers all the vertices of the given graph. The celebrated theorem of Hajnal and Szemerédi asserts that every graph on n vertices with minimum degree at least (1-1/r)n contains a K_r-factor. In this note, we propose investigating the relation between minimum degree and existence of perfect K_r-packing for edge-weighted graphs. The main question we study is the following. Suppose that a positive integer r>=2 and a real t in [0,1] is given. What is the minimum weighted degree of K_n that guar…
An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer $k$, there exists a constant $c_k>0$ such that any ordered graph $G$ on $n$ vertices with the property that neither $G$ nor its complement contains an induced monotone path of size $k$, has either a clique or an independent set of size at least $n^{c_k}$. This strengthens a result of Bousquet, Lagoutte, and Thomassé, who proved the analogous result for unordered graphs. A key idea of the above paper was to show that any unordered graph on $n$ vertices that does not contain an induced …
The celebrated Erdös-Hajnal conjecture states that for every undirected graph $H$ there exists $ ε(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $ n^{ε(H)} $. This conjecture has a directed equivalent version stating that for every tournament $H$ there exists $ ε(H) > 0 $ such that every $H$-free $n$-vertex tournament $T$ contains a transitive subtournament of order at least $ n^{ε(H)} $. This conjecture is proved for few infinite families of tournaments. In this paper we construct…
A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph $H$ there exists $ ε(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $ n^{ε(H)} $. The conjecture has a directed equivalent version stating that for every tournament $H$ there exists $ ε(H) > 0 $ such that every $H-$free $n-$vertex tournament $T$ contains a transitive subtournament of order at least $ n^{ε(H)} $. Both the directed and the undirected versions of the conjecture are known to be…
We say that a (di)graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. The seminal Hajnal--Szemerédi theorem characterises the minimum degree that ensures a graph $G$ contains a perfect $K_r$-packing. In this paper we prove the following analogue for directed graphs: Suppose that $T$ is a tournament on $r$ vertices and $G$ is a digraph of sufficiently large order $n$ where $r$ divides $n$. If $G$ has minimum in- and outdegree at least $ (1-1/r)n$ then $G$ contains a perfect $T$-packing. In the case when $T$ is a cyclic …
Let $q$ be a positve integer, and $G$ be a $q$-partite simple graph on $qn$ vertices, with $n$ vertices in each vertex class. Let $δ={k_q \over k_q+1}$, where $k_q=q+O(\log{q})$. If each vertex of $G$ is adjacent to at least $δn$ vertices in each of the other vertex classes, $q$ is bounded and $n$ is large enough, then $G$ has a $K_q$-factor.
Erdös-Hajnal conjecture states that for every undirected graph $H$ there exists $ ε(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $ n^{ε(H)} $. This conjecture has a directed equivalent version stating that for every tournament $H$ there exists $ ε(H) > 0 $ such that every $H-$free $n-$vertex tournament $T$ contains a transitive subtournament of order at least $ n^{ε(H)} $. This conjecture is known to hold for a few infinite families of tournaments. In this paper we construct two n…
Erdős and Hajnal conjectured that, for every graph $H$, there exists a constant $c_H$ such that every graph $G$ on $n$ vertices which does not contain any induced copy of $H$ has a clique or a stable set of size $n^{c_H}$. We prove that for every $k$, there exists $c_k>0$ such that every graph $G$ on $n$ vertices not inducing a cycle of length at least $k$ nor its complement contains a clique or a stable set of size $n^{c_k}$.
The celebrated Erdős-Hajnal conjecture states that for every proper hereditary graph class $\mathcal{G}$ there exists a constant $\varepsilon = \varepsilon(\mathcal{G}) > 0$ such that every graph $G \in \mathcal{G}$ contains a clique or an independent set of size $|V(G)|^\varepsilon$. Recently, there has been a growing interest in the symmetrized variant of this conjecture, where one additionally requires $\mathcal{G}$ to be closed under complementation. We show that any hereditary graph class that is closed under complementation and excludes a fixed caterpillar as an induced subgraph satisfie…
A seminal result of Hajnal and Szemerédi states that if a graph $G$ with $n$ vertices has minimum degree $δ(G) \ge (r-1)n/r$ for some integer $r \ge 2$, then $G$ contains a $K_r$-factor, assuming $r$ divides $n$. Extremal examples which show optimality of the bound on $δ(G)$ are very structured and, in particular, contain large independent sets. In analogy to the Ramsey-Turán theory, Balogh, Molla, and Sharifzadeh initiated the study of how the absence of such large independent sets influences sufficient minimum degree. We show the following two related results: $\bullet$ For any $r > \ell \ge…
Let $G$ be a $k$-partite graph with $n$ vertices in parts such that each vertex is adjacent to at least $δ^*(G)$ vertices in each of the other parts. Magyar and Martin \cite{MaMa} proved that for $k=3$, if $δ^*(G)\ge 2/3n $ and $n$ is sufficiently large, then $G$ contains a $K_3$-factor (a spanning subgraph consisting of $n$ vertex-disjoint copies of $K_3$) except that $G$ is one particular graph. Martin and Szemerédi \cite{MaSz} proved that $G$ contains a $K_4$-factor when $δ^*(G)\ge 3/4n$ and $n$ is sufficiently large. Both results were proved by the Regularity Lemma. In this paper we give a…
Erdős and Szekeres's quantitative version of Ramsey's theorem asserts that any complete graph on n vertices that is edge-colored with two colors has a monochromatic clique on at least 1/2log(n) vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed color patterns ensures larger monochromatic cliques. Specifically, it claims that for any fixed integer k and any clique K on k vertices edge-colored with two colors, there is a positive constant a such that in any complete n-vertex graph edge-colored with two colors that does not contain a copy of K, there is a monochromatic cli…
Hajós conjecture asserts that a simple Eulerian graph on n vertices can be decomposed into at most (n - 1)/2 cycles. The conjecture is only proved for graph classes in which every element contains vertices of degree 2 or 4. We develop new techniques to construct cycle decompositions. They work on the common neighbourhood of two degree-6 vertices. With these techniques we find structures that cannot occur in a minimal counterexample to Hajós conjecture and verify the conjecture for Eulerian graphs of pathwidth at most 6. This implies that these graphs satisfy the small cycle double cover conjec…
For $k \in \mathbb N$, Corrádi and Hajnal proved that every graph $G$ on $3k$ vertices with minimum degree $δ(G) \ge 2k$ has a $C_3$-factor, i.e., a partitioning of the vertex set so that each part induces the 3-cycle $C_3$. Wang proved that every directed graph $\overrightarrow G$ on $3k$ vertices with minimum total degree $δ_t(\overrightarrow G):=\min_{v\in V}(deg^-(v)+deg^+(v)) \ge 3(3k-1)/2$ has a $\overrightarrow C_3$-factor, where $\overrightarrow C_3$ is the directed 3-cycle. The degree bound in Wang's result is tight. However, our main result implies that for all integers $a \ge 1$ and…
We prove that for every k, there exists $c_k>0$ such that every graph G on n vertices not inducing a path $P_k$ and its complement contains a clique or a stable set of size $n^{c_k}$.
The chromatic number $\overrightarrowχ(D)$ of a digraph $D$ is the minimum number of colors needed to color the vertices of $D$ such that each color class induces an acyclic subdigraph of $D$. A digraph $D$ is $k$-critical if $\overrightarrowχ(D) = k$ but $\overrightarrowχ(D') < k$ for all proper subdigraphs $D'$ of $D$. We examine methods for creating infinite families of critical digraphs, the Dirac join and the directed and bidirected Hajós join. We prove that a digraph $D$ has chromatic number at least $k$ if and only if it contains a subdigraph that can be obtained from bidirected complet…
The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free, that is, it does not contain an induced copy of a given graph H, then it must contain either a clique or an independent set of size n^{d(H)}, where d(H) > 0 depends only on the graph H. Except for a few special cases, this conjecture remains wide open. However, it is known that a H-free graph must contain a complete or empty bipartite graph with parts of polynomial size. We prove an analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform hypergraph on n vertices is H-free, for any given H, then i…
Hajós conjectured that every graph containing no subdivision of the complete graph $K_{s+1}$ is properly $s$-colorable. This conjecture was disproved by Catlin. Indeed, the maximum chromatic number of such graphs is $Ω(s^2/\log s)$. We prove that $O(s)$ colors are enough for a weakening of this conjecture that only requires every monochromatic component to have bounded size (so-called clustered coloring). Our approach leads to more results. Say that a graph is an almost $(\leq 1)$-subdivision of a graph $H$ if it can be obtained from $H$ by subdividing edges, where at most one edge is subdivid…
The following natural problem was raised independently by Erdős-Hajnal and Linial-Rabinovich in the late 80's. How large must the independence number $α(G)$ of a graph $G$ be whose every $m$ vertices contain an independent set of size $r$? In this paper we discuss new methods to attack this problem. The first new approach, based on bounding Ramsey numbers of certain graphs, allows us to improve previously best lower bounds due to Linial-Rabinovich, Erdős-Hajnal and Alon-Sudakov. As an example, we prove that any $n$-vertex graph $G$ having an independent set of size $3$ among every $7$ vertices…
London School of Economics and Political Science (LSE)（伦敦政治经济学院）；Colorado State University（科罗拉多州立大学）；King’s College London（伦敦国王学院）；Universitat Politècnica de Catalunya (UPC)（加泰罗尼亚理工大学）
The Erdős-Hajnal Conjecture states that for every $H$ there exists a constant $ε(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $|V(G)|^{ε(H)}$. The Conjecture is still open. Some time ago its directed version was formulated (see:\cite{alon}). In the directed version graphs are replaced by tournaments, and cliques and stable sets by transitive subtournaments. If the Conjecture is not true then the smallest counterexample is a prime tournament. For a long time the Conjecture was known only for finitely many pri…
Inspired by a question of Yannakakis on the Vertex Packing polytope of perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates a clique K and a stable set S if $K\subseteq B$ and $S\subseteq W$. A Clique-Stable Set Separator is a family of cuts such that for every clique K, and for every stable set S disjoint from K, there exists a cut in the family that separates K and S. Given a class of graphs, the question is to know whether every graph of the class admits a Clique-Stable Set Separator cont…
Hadwiger and Hajós conjectured that for every positive integer $t$, $K_{t+1}$-minor free graphs and $K_{t+1}$-topological minor free graphs are properly $t$-colorable, respectively. Clustered coloring version of these two conjectures which only require monochromatic components to have bounded size has been extensively studied. In this paper we consider the clustered coloring version of the immersion-variant of Hadwiger's and Hajós' conjecture proposed by Lescure and Meyniel and independently by Abu-Khzam and Langston. We determine the minimum number of required colors for $H$-immersion free gr…
