Dong Yeap Kang, Tom Kelly, Daniela Kühn, Abhishek Methuku, Deryk Osthus
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.
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Generalizing the concept of dense hypergraph, we say that a hypergraph is weakly dense, if no k in the half-open interval [2,sqrt(n)) is the degree of more than k^2 vertices. In our main result, we prove the famous Erdos-Faber-Lovasz conjecture when the hypergraph is weakly dense.
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 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…
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 …
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 $\ell=0$.
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 …
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 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…
An Improved Lower Bound for the Erdős-Lovász Cover Number Problem
本文通过初等方法证明$g(r)\ge 3r-4$，并利用Kahn的小余度超图边着色定理改进为$g(r)\ge (61/20-o(1))r$，这是约五十年来首次改进Erdős-Lovász下界。
Some Cases of the Erdős-Lovász Tihany Conjecture for Claw-free Graphs
本文针对无爪图研究 Erdős-Lovász Tihany 猜想，在含有 $K_s$ 的条件下分别证明了 $t \leq s+2$ 和 $t \leq 4s-3$ 两类情形成立，并证明了无爪图在数对 $(3,10)$ 上的猜想。
A resolution of Erdős Problem #190 via Erdős-Lovász, BCT, and Baker-Harman-Pintz
本文利用Erdős-Lovász、BCT和Baker-Harman-Pintz定理，证明了H(k)^{1/k}/k随着k增长而趋于无穷，解决了Erdős-Graham问题的正向方向。
The Lovász-Cherkassky theorem in infinite graphs
本文在无限图背景下，将Lovász-Cherkassky定理推广到“内欧拉”设置中边不相交$ T $-路径的打包问题，延续Erdős-Menger猜想的无限推广思路。
The Erdős-Lovász Tihany Conjecture holds for all even-hole-free graphs
研究满足特定条件的图的可拆分性，核心方法是证明不含\(C_4\)且每个诱导子图有双单纯顶点的图是\((s,t)\)-可拆分的，结合相关结果，主要贡献是证明埃尔德什 - 洛瓦兹蒂哈尼猜想对所有不含偶洞的图成立。
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 …
Motivated by the Erdos-Faber Lovasz conjecture (EFL) for hypergraphs, we explore relationships between several conjectures on the edge coloring of linear hypergraphs. In particular, we are able to increase the class of hypergraphs for which EFL is true.
Defect and transference versions of the Alon-Frankl-Lovasz theorem
本文提出了阿隆-弗兰克尔-洛瓦茨定理的缺陷和传递版本，通过随机超图的染色分析，证明了在特定条件下存在近似大小的单色匹配。
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.
On an Erdős--Lov'asz problem: 3-critical 3-graphs of minimum degree 7
本文解决了埃尔多斯-洛瓦茨问题，构造了一个3-均匀超图，其最小度数为7且满足特定临界条件。
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, …
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.
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 $…
Motivated by the Erdős-Faber-Lovász (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We discuss several conjectures for list edge coloring linear hypergraphs that generalize both EFL and Vizing's theorem for graphs. For example, we conjecture that in a linear hypergraph of rank 3, the list edge chromatic number is at most 2 times the maximum degree plus 1. We show that for sufficiently large fixed rank and sufficiently large degree, the conjectures are true.
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…
The Berge-Füredi conjecture on the chromatic index of hypergraphs with large hyperedges
本文围绕与超图着色相关的两个猜想（Vizing定理的超图推广、Erdős-Faber-Lovász猜想），研究一类特殊均匀线性超图并建立其若干性质，推进Berge-Füredi猜想相关研究。
About Berge-Füredi's conjecture on the chromatic index of hypergraphs
本文在反秩或最大度的特定条件下证明了超图边染色数的Berge-Füredi猜想上界，并为Erdős-Faber-Lovász猜想提供了精确信息。
A note on extremal constructions for the Erdős--Rademacher problem
本文针对Erdős–Rademacher问题，显式给出了Lovász与Simonovits猜想预测的$r$-团最小数量，并修正了Pikhurko与Razborov的猜想，提出$r\ge 4$且$n$充分大时的极值图集合。
Edge-coloring linear hypergraphs with medium-sized edges
针对线性超图，受EFL猜想启发，证明了当边大小在$i$到$C_{i,ε}\sqrt{n}$之间时，可用$(1+ε)n/(i-1)$种颜色进行列表边染色，且对$n$的依赖最优。
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 $6$.
Let $ H = (V,E) $ be a hypergraph. By the chromatic number of a hypergraph $ H = (V,E) $ we mean the minimum number $χ(H)$ of colors needed to paint all the vertices in $ V $ so that any edge $ e \in E $ contains at least two vertices of some different colors. Finally, a hypergraph is said to form a clique, if its edges are pairwise intersecting. In 1973 Erdős and Lovász noticed that if an $n$-uniform hypergraph $ H = (V,E) $ forms a clique, then $ χ(H) \in \{2,3\} $. They untoduced following quantity. $$ M(n) = \max \{|E|: \exists {\rm an} n-{\rm uniform} {\rm clique} H = (V,E) {\rm with} χ(H…
Solution to a problem of Erdős on the chromatic index of hypergraphs with bounded codegree
本文解决了 Erdős 关于有界共度超图色指标的问题，证明最大度至多为 (1-o(1))tn 且最大共度至多为 t 的 n 顶点超图色指标至多为 tn，并刻画了取等时的极值结构。
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>…
A decomposition of a non-empty simple graph $G$ is a pair $[G,P]$, such that $P$ is a set of non-empty induced subgraphs of $G$, and every edge of $G$ belongs to exactly one subgraph in $P$. The chromatic index $χ'([G,P])$ of a decomposition $[G,P]$ is the smallest number $k$ for which there exists a $k$-coloring of the elements of $P$ in such a way that: for every element of $P$ all of its edges have the same color, and if two members of $P$ share at least one vertex, then they have different colors. A long standing conjecture of Erdős-Faber-Lovász states that every decomposition $[K_n,P]$ of…
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…
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…
An Erdős Matching Conjecture for Vector Spaces
研究向量空间中 Erdős 匹配猜想的类比，通过构造和组合方法证明了对 k=2、n=(s+1)k 及 n 足够大时猜想成立，并得到稳定性定理和极值数结果。
This note concerns a well-known result which we term the ``spread lemma,'' which establishes the existence (with high probability) of a desired structure in a random set. The spread lemma was central to two recent celebrated results: (a) the improved bounds of Alweiss, Lovett, Wu, and Zhang (2019) on the Erdős-Rado sunflower conjecture; and (b) the proof of the fractional Kahn--Kalai conjecture by Frankston, Kahn, Narayanan and Park (2019). While the lemma was first proved (and later refined) by delicate counting arguments, alternative proofs have also been given, via Shannon's noiseless codin…
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