Let $s\ge2$ and $t\ge2$ be integers. A graph $G$ is $(s,t)$-\emph{splittable} if $V(G)$ can be partitioned into two sets $S$ and $T$ such that $χ(G[S])\geq s$ and $χ(G[T])\geq t$. The well-known Erdős-Lovász Tihany Conjecture from 1968 states that every graph $G$ whose chromatic number $χ(G)=s+t-1$ is more than its clique number $ω(G)$ is $(s,t)$-splittable. In this paper, we prove an enhanced version of the Erdős-Lovász Tihany Conjecture for graphs with independence number two. That is, for every graph $G$ with $χ(G)=s+t-1>ω(G)+1$ is $(s,t+1)$-splittable. There are examples showing that this result is best possible.
In this paper, we prove an enhanced version of the Erdős-Lovász Tihany Conjecture for line graphs of multigraphs. That is, for every graph $G$ whose chromatic number $χ(G)$ is more than its clique number $ω(G)$ and for nonnegative integer $\ell$, any two integers $s,t \geq 3.5\ell+2$ with $s+t = χ(G)+1$, there is a partition $(S,T)$ of the vertex set $V(G)$ such that $χ(G[S])\geq s$ and $χ(G[T])\geq t+\ell$. In particular, when $\ell=1$, we can obtain the same result just for any $s,t\geq4$. The Erdős-Lovász Tihany conjecture is a special case when $\ell=0$.
A hole in a graph is an induced cycle of length at least $4$. Let $s\ge2$ and $t\ge2$ be integers. A graph $G$ is $(s,t)$-splittable if $V(G)$ can be partitioned into two sets $S$ and $T$ such that $χ(G[S ]) \ge s$ and $χ(G[T ]) \ge t$. The well-known Erdős-Lovász Tihany Conjecture from 1968 states that every graph $G$ with $ω(G) < χ(G) = s + t - 1$ is $(s,t)$-splittable. This conjecture is hard, and few related results are known. However, it has been verified to be true for line graphs, quasi-line graphs, and graphs with independence number $2$. In this paper, we establish more evidence for t…
In 1968, Erdös and Lovász conjectured that for every graph $G$ and all integers $s,t\geq 2$ such that $s+t-1=χ(G) > ω(G)$, there exists a partition $(S,T)$ of the vertex set of $G$ such that $χ(G|S)\geq s$ and $χ(G|T)\geq t$. For general graphs, the only settled cases of the conjecture are when $s$ and $t$ are small. Recently, the conjecture was proved for a few special classes of graphs: graphs with stability number 2 \cite{quasi-line}, line graphs \cite{line} and quasi-line graphs \cite{quasi-line}. In this paper, we consider the conjecture for claw-free graphs and present some progress on i…
A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph has a bisimplicial vertex. Both Hadwiger's conjecture and the Erdős-Lovász Tihany conjecture have been shown to be true for quasi-line graphs, but are open for even-hole-free graphs. In this note, we prove that for all $k\ge7$, every even-hole-free graph with no $K_k$ minor is $(2k-5)$-colorable; every even-hole-free graph $G$ with $ω(G)<χ(G)=s+t-1$ satisfie…
We consider the Erd{ő}s - Faber - Lovász (EFL) conjecture for hypergraphs. This paper gives an upper bound for the chromatic number of $r$ regular linear hypergraphs $\textbf{H}$ of size $n$. If $r \ge 4$, $χ(\textbf{H}) \le 1.181n$ and if $r=3$, $χ(\textbf{H}) \le 1.281n$
The Erdős-Faber-Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In this paper, we prove this conjecture for every large $n$. We also provide stability versions of this result, which confirm a prediction of Kahn.
The long-standing Erdős-Faber-Lovász conjecture states that every $n$-uniform linear hypergaph with $n$ edges has a proper vertex-coloring using $n$ colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdős-Faber-Lovász conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations,…
A linear hypergraph is intersecting if any two different edges have exactly one common vertex and an $n$-quasicluster is an intersecting linear hypergraph with $n$ edges each one containing at most $n$ vertices and every vertex is contained in at least two edges. The Erdös-Faber-Lovász Conjecture states that the chromatic number of any $n$-quasicluster is at most $n$. In the present note we prove the correctness of the conjecture for a new infinite class of $n$-quasiclusters using a specific edge coloring of the complete graph.
The Erdős--Faber--Lovász Conjecture, posed in 1972, states that if a graph $G$ is the union of $n$ cliques of order $n$ (referred to as defining $n$-cliques) such that two cliques can share at most one vertex, then the vertices of $G$ can be properly coloured using $n$ colours. Although still open after almost 50 years, it can be easily shown that the conjecture is true when every shared vertex belongs to exactly two defining $n$-cliques. We here provide a quick and easy algorithm to colour the vertices of $G$ in this case, and discuss connections with clique-decompositions and edge-colourings…
In 1972, Erdös - Faber - Lovász (EFL) conjectured that, if $\textbf{H}$ is a linear hypergraph consisting of $n$ edges of cardinality $n$, then it is possible to color the vertices with $n$ colors so that no two vertices with the same color are in the same edge. In 1978, Deza, Erdös and Frankl had given an equivalent version of the same for graphs: Let $G= \bigcup _{i=1}^{n} A_i$ denote a graph with $n$ complete graphs $A_1, A_2,$ $ \dots , A_n$, each having exactly $n$ vertices and have the property that every pair of complete graphs has at most one common vertex, then the chromatic number of…
A decomposition of a simple graph $G$ is a pair $(G,P)$ where $P$ is a set of subgraphs of $G$, which partitions the edges of $G$ in the sense that every edge of $G$ belongs to exactly one subgraph in $P$. If the elements of $P$ are induced subgraphs then the decomposition is denoted by $[G,P]$. A $k$-$P$-coloring of a decomposition $(G,P)$ is a surjective function that assigns to the edges of $G$ a color from a $k$-set of colors, such that all edges of $H\in P$ have the same color, and, if $H_1,H_2\in P$ with $V(H_1)\cap V(H_2)\neq\emptyset$ then $E(H_1)$ and $E(H_2)$ have different colors. T…
Erdős and Lovász noticed that an $r$-uniform intersecting hypergraph $H$ with maximal covering number, that is $τ(H)=r$, must have at least $\frac{8}{3}r-3$ edges. There has been no improvement on this lower bound for 45 years. We try to understand the reason by studying some small cases to see whether the truth lies very close to this simple bound. Let $q(r)$ denote the minimum number of edges in an intersecting $r$-uniform hypergraph. It was known that $q(3)=6$ and $q(4)=9$. We obtain the following new results: The extremal example for uniformity 4 is unique. Somewhat surprisingly it is not …
In 1972, Erdös - Faber - Lovász (EFL) conjectured that, if $\textbf{H}$ is a linear hypergraph consisting of $n$ edges of cardinality $n$, then it is possible to color the vertices with $n$ colors so that no two vertices with the same color are in the same edge. In 1978, Deza, Erdös and Frankl had given an equivalent version of the same for graphs: Let $G= \bigcup_{i=1}^{n} A_i$ denote a graph with $n$ complete graphs $A_1, A_2,$ $ \dots , A_n$, each having exactly $n$ vertices and have the property that every pair of complete graphs has at most one common vertex, then the chromatic number of …
For a hypergraph $H$, define its intersection spectrum $I(H)$ as the set of all intersection sizes $|E\cap F|$ of distinct edges $E,F\in E(H)$. In their seminal paper from 1973 which introduced the local lemma, Erdős and Lovász asked: how large must the intersection spectrum of a $k$-uniform $3$-chromatic intersecting hypergraph be? They showed that such a hypergraph must have at least three intersection sizes, and conjectured that the size of the intersection spectrum tends to infinity with $k$. Despite the problem being reiterated several times over the years by Erdős and other researchers, …
We apply Fourier analysis on finite groups to obtain simplified formulations for the Lovász theta-number of a Cayley graph. We put these formulations to use by checking a few cases of a conjecture of Ellis, Friedgut, and Pilpel made in a recent article proving a version of the Erdős-Ko-Rado theorem for $k$-intersecting families of permutations. We also introduce a $q$-analog of the notion of $k$-intersecting families of permutations, and we verify a few cases of the corresponding Erdős-Ko-Rado assertion by computer.
Lovász and Cherkassky discovered in the 1970s independently that if $ G $ is a finite graph with a given set $ T $ of terminal vertices such that $ G $ is inner Eulerian, then the maximal number of edge-disjoint paths connecting distinct vertices in $ T $ is $ \sum_{t\in T}λ(t, T-t) $ where $λ$ is the local edge-connectivity function. The optimality of a system of edge-disjoint $ T $-paths in the Lovász-Cherkassky theorem is witnessed by the existence of certain cuts by Menger's theorem. The infinite generalisation of Menger's theorem by Aharoni and Berger (earlier known as the Erdős-Menger Co…
The triangle covering number of a graph is the minimum number of vertices that hit all triangles. Given positive integers $s,t$ and an $n$-vertex graph $G$ with $\lfloor n^2/4 \rfloor +t$ edges and triangle covering number $s$, we determine (for large $n$) sharp bounds on the minimum number of triangles in $G$ and also describe the extremal constructions. Similar results are proved for cliques of larger size and color critical graphs. This extends classical work of Rademacher, Erd\H os, and Lovász-Simonovits whose results apply only to $s \le t$. Our results also address two conjectures of Xia…
It follows from a theorem of Lovász that if $ D $ is a finite digraph with $ r\in V(D) $ then there is a spanning subdigraph $ E $ of $ D $ such that for every vertex $ v\neq r $ the following quantities are equal: the local connectivity from $ r $ to $ v $ in $ D $, the local connectivity from $ r $ to $ v $ in $ E $ and the indegree of $ v $ in $ E $. In infinite combinatorics cardinality is often an overly rough measure to obtain deep results and it is more fruitful to capture structural properties instead of just equalities between certain quantities. The best known example for such a resu…
In 1973 P. Erdős and L. Lovász noticed that any hypergraph whose edges are pairwise intersecting has chromatic number 2 or 3. In the first case, such hypergraph may have any number of edges. However, Erdős and Lovász proved that in the second case, the number of edges is bounded from above. For example, if a hypergraph is $ n $-uniform, has pairwise intersecting edges, and has chromatic number 3, then the number of its edges does not exceed $ n^n $. Recently D.D. Cherkashin improved this bound (see \cite{Ch}). In this paper, we further improve it in the case when the number of vertices of an $…
Mantel's theorem states that every $n$-vertex graph with $\lfloor \frac{n^2}{4} \rfloor +t$ edges, where $t>0$, contains a triangle. The problem of determining the minimum number of triangles in such a graph is usually referred to as the Erdős-Rademacher problem. Lovász and Simonovits proved that there are at least $t\lfloor n/2 \rfloor$ triangles in each of those graphs. Katona and Xiao considered the same problem under the additional condition that there are no $s-1$ vertices covering all triangles. They settled the case $t=1$ and $s=2$. Solving their conjecture, we determine the minimum num…
Alon, Frankl, and Lovász proved a conjecture of Erdős that one needs at least $\lceil \frac{n-r(k-1)}{r-1} \rceil$ colors to color the $k$-subsets of $\{1, \dots, n\}$ such that any $r$ of the $k$-subsets that have the same color are not pairwise disjoint. A generalization of this problem where one requires $s$-wise instead of pairwise intersections was considered by Sarkaria. He claimed a proof of a generalized Erdős--Kneser conjecture establishing a lower bound for the number of colors that reduces to Erdős' original conjecture for ${s = 2}$. Lange and Ziegler pointed out that his proof fail…
Paul Erdős and László Lovász proved in a landmark article that, for any positive integer $k$, up to isomorphism there are only finitely many maximal intersecting families of $k-$sets (maximal $k-$cliques). So they posed the problem of determining or estimating the largest number $N(k)$ of the points in such a family. They also proved by means of an example that $N(k)\geq2k-2+\frac{1}{2}\binom{2k-2}{k-1}$. Much later, Zsolt Tuza proved that the bound is best possible up to a multiplicative constant by showing that asymptotically $N(k)$ is at most $4$ times this lower bound. In this paper we red…
A connected $t$-chromatic graph $G$ is \dfn{double-critical} if $G \backslash\{u, v\}$ is $(t-2)$-colorable for each edge $uv\in E(G)$. A long standing conjecture of Erdős and Lovász that the complete graphs are the only double-critical $t$-chromatic graphs remains open for all $t\ge6$. Given the difficulty in settling Erdős and Lovász's conjecture and motivated by the well-known Hadwiger's conjecture, Kawarabayashi, Pedersen and Toft proposed a weaker conjecture that every double-critical $t$-chromatic graph contains a $K_t$ minor and verified their conjecture for $t\le7$. Albar and Gonçalves…
Our previous paper applied a lopsided version of the Lovász Local Lemma that allows negative dependency graphs to the space of random injections from an $m$-element set to an $n$-element set. Equivalently, the same story can be told about the space of random matchings in $K_{n,m}$. Now we show how the cited version of the Lovász Local Lemma applies to the space of random matchings in $K_{2n}$. We also prove tight upper bounds that asymptotically match the lower bound given by the Lovász Local Lemma. As a consequence, we give new proofs to results on the enumeration of $d$-regular graphs. The t…
Paul Erdős and László Lovász established by means of an example that there exists a maximal intersecting family of $k-$sets with approximately $(e-1)k!$ blocks. László Lovász conjectured that their example is best known example which has the maximum number of blocks. Later it was disproved. But the quest for such examples remain valid till this date. In this short note, by computing transversal size of a certain series of uniform intersecting families constructed over the cycle graph, we provide an example which has more than $(\frac{k}{2})^{k-1}$ (approximately) blocks.
Let $x_1, \ldots, x_n \in \mathbb{R}^d$ be unit vectors such that among any three there is an orthogonal pair. How large can $n$ be as a function of $d$, and how large can the length of $x_1 + \ldots + x_n$ be? The answers to these two celebrated questions, asked by Erdős and Lovász, are closely related to orthonormal representations of triangle-free graphs, in particular to their Lovász $\vartheta$-function and minimum semidefinite rank. In this paper, we study these parameters for general $H$-free graphs. In particular, we show that for certain bipartite graphs $H$, there is a connection bet…
A connected $n$-chromatic graph $G$ is double-critical if for all the edges $xy$ of $G$, the graph $G-x-y$ is $(n-2)$-chromatic. In 1966, Erd\H os and Lovász conjectured that the only double-critical $n$-chromatic graph is $K_n$. This conjecture remains unresolved for $n \ge 6.$ In this short note, we verify this conjecture for claw-free graphs $G$ of chromatic number $6$.
What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first non-trivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős-Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from~$1$, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we gi…
In this paper, we present a minimal chordal completion $G^*$ of a graph $G$ satisfying the inequality $ω(G^*) - ω(G) \le i(G)$ for the non-chordality index $i(G)$ of $G$. In terms of our chordal completions, we partially settle the Hadwiger conjecture and the Erdős-Faber-Lovász Conjecture, and extend the known $χ$-bounded class by adding to it the family of graphs with bounded non-chordality indices.
What does an Erdos-Renyi graph look like when a rare event happens? This paper answers this question when p is fixed and n tends to infinity by establishing a large deviation principle under an appropriate topology. The formulation and proof of the main result uses the recent development of the theory of graph limits by Lovasz and coauthors and Szemeredi's regularity lemma from graph theory. As a basic application of the general principle, we work out large deviations for the number of triangles in G(n,p). Surprisingly, even this simple example yields an interesting double phase transition.
The celebrated Erdos, Faber and Lovasz conjecture may be stated as follows: Any linear hypergraph on v points has chromatic index at most v. We will introduce the linear intersection number of a graph, and use this number to give an alternative formulation of the conjecture. Finally, first results about the linear intersection number will be proved. For example, we will determine all graphs with maximal linear intersection number given the number of edges of the graph.
Hajos' conjecture that every simple even graph on $n$ vertices can be decomposed into at most $(n-1)/2$ cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp. 231 - 236). Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices in which no two cycles have the same length. P. Erdos raised the problem of determining $f(n)$ (see J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, New York, 1976), p.247, Problem 11). Given a graph $H$, what is the maximum number of edges of a graph with…
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…
Let ${n_k}$ be an increasing lacunary sequence, i.e., $n_{k+1}/n_k>1+r$ for some $r>0$. In 1987, P. Erdos asked for the chromatic number of a graph $G$ on the integers, where two integers $a,b$ are connected by an edge iff their difference $|a-b|$ is in the sequence ${n_k}$. Y. Katznelson found a connection to a Diophantine approximation problem (also due to Erdos): the existence of $x$ in $(0,1)$ such that all the multiples $n_j x$ are at least distance $δ(x)>0$ from the set of integers. Katznelson bounded the chromatic number of $G$ by $Cr^{-2}|\log r|$. We apply the Lovász local lemma to es…
In 1976 Erdos, Kleitman and Rothschild determined the number of graphs without a clique of size $\ell$. In this note we extend their result to the case of forbidden cliques of increasing size. More precisely we prove that for $\ell_n \le \frac12(\log n)^{1/4}$ there are $$2^{(1-1/(\ell_n-1))n^2/2+o(n^2/\ell_n)}$$ $K_{\ell_n}$-free graphs of order $n$. Our proof is based on the recent hypergraph container theorems of Saxton, Thomason and Balogh, Morris, Samotij, in combination with a theorem of Lovasz and Simonovits.
Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；School of Mathematics and Statistics, Central South University（中南大学数学与统计学院）；School of Mathematics, Hunan University（湖南大学数学学院）
We prove that $s_r(K_k) = O(k^5 r^{5/2})$, where $s_r(K_k)$ is the Ramsey parameter introduced by Burr, Erdős and Lovász in 1976, which is defined as the smallest minimum degree of a graph $G$ such that any $r$-colouring of the edges of $G$ contains a monochromatic $K_k$, whereas no proper subgraph of $G$ has this property. The construction used in our proof relies on a group theoretic model of generalised quadrangles introduced by Kantor in 1980.
A connected $k$-chromatic graph $G$ is double-critical if for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. The only known double-critical $k$-chromatic graph is the complete $k$-graph $K_k$. The conjecture that there are no other double-critical graphs is a special case of a conjecture from 1966, due to Erdős and Lovász. The conjecture has been verified for $k \leq 5$. We prove for $k=6$ and $k=7$ that any non-complete double-critical $k$-chromatic graph is 6-connected and has $K_k$ as a minor.
In this paper, we prove a generalization of a conjecture of Erdös, about the chromatic number of certain Kneser-type hypergraphs. For integers $n,k,r,s$ with $n\ge rk$ and $2\le s\le r$, the $r$-uniform general Kneser hypergraph $\mbox{KG}^r_s(n,k)$, has all $k$-subsets of $\{1,\dots,n\}$ as the vertex set and all multi-sets $\{A_1,\dots, A_r\}$ of $k$-subsets with $s$-wise empty intersections as the edge set. The case $r=s=2$, was considers by Kneser \cite{K} in 1955, where he conjectured that its chromatic number is $n-2(k-1)$. This was finally proved by Lovász \cite{L} in 1978. The case $r>…
We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying planar graphs are Halin graphs, cactus graphs, maximal planar graphs, and triangle-free planar graphs, as well as randomly produced segment sets. We also apply these results to a variant of the Erdős-Faber-Lovász (EFL) Conjecture stating that the intersection points of $m$ segments can be colored with $m$ colors so that no segment contains points with the s…
Let $F$ be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of $F$ in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits, who proved that there is one copy of $F$, and of Rademacher, Erd\H os and Lovász-Simonovits, who proved similar counting results when $F$ is a complete graph. One of the simplest cases of our theorem is the following new result. There is an absolute positive constant $c$ such that if $n$ is sufficiently large and $1 \le q < cn$, then every $n$ vertex graph wi…
A connected graph $G$ with chromatic number $t$ is double-critical if $G \backslash \{x, y\}$ is $(t - 2)$-colorable for each edge $xy \in E(G)$. The complete graphs are the only known examples of double-critical graphs. A long-standing conjecture of Erd\H os and Lovász from 1966, which is referred to as the Double-Critical Graph Conjecture, states that there are no other double-critical graphs. That is, if a graph $G$ with chromatic number $t$ is double-critical, then $G$ is the complete graph on $t$ vertices. This has been verified for $t \le 5$, but remains open for $t \ge 6$. In this paper…
This article is built upon three main ideas. First, for a class of monomial ideals, it is proven that the multiplicity of an ideal equals the number of realizations of its codimension (an intuitive concept that we define later). Next, for an arbitrary graph G, we construct a monomial ideal M_G, and show that the chromatic number of G is equal to the codimension of M_G. Finally, for a class of graphs, we give a formula that computes the chromatic polynomial of G, evaluated at the chromatic number of G, in terms of the codimension and multiplicity of M_G. In particular, the formula applies to al…
We prove that in sparse Erdős-Rényi graphs of average degree $d$, the vector chromatic number (the relaxation of chromatic number coming from the Lovàsz theta function) is typically $\tfrac{1}{2}\sqrt{d} + o_d(1)$. This fits with a long-standing conjecture that various refutation and hypothesis-testing problems concerning $k$-colorings of sparse Erdős-Rényi graphs become computationally intractable below the `Kesten-Stigum threshold' $d_{KS,k} = (k-1)^2$. Along the way, we use the celebrated Ihara-Bass identity and a carefully constructed non-backtracking random walk to prove two deterministic…
The \emph{Turán function} $\ex(n,F)$ of a graph $F$ is the maximum number of edges in an $F$-free graph with $n$ vertices. The classical results of Turán and Rademacher from 1941 led to the study of supersaturated graphs where the key question is to determine $h_F(n,q)$, the minimum number of copies of $F$ that a graph with $n$ vertices and $\ex(n,F)+q$ edges can have. We determine $h_F(n,q)$ asymptotically when $F$ is \emph{color-critical} (that is, $F$ contains an edge whose deletion reduces its chromatic number) and $q=o(n^2)$. Determining the exact value of $h_F(n,q)$ seems rather difficul…
