相关论文: Bounds on Erd{\H{o}}s - Faber - Lov\'{a}sz Conject…
Motivated by the Erd\H{o}s-Faber-Lov\'asz (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…
The Erd\H{o}s-Faber-Lov\'{a}sz 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…
Motivated by the Erd\H{o}s-Faber-Lov\'{a}sz (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We show that if the hyper-edge sizes are bounded between $i$ and $C_{i,\epsilon} \sqrt{n}$ inclusive,…
In 1972, Erd\"{o}s - Faber - Lov\'{a}sz (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…
In 1972, Erd\"{o}s - Faber - Lov\'{a}sz (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…
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…
We show that the chromatic index of a hypergraph $\mathcal{H}$ satisfies Berge-F\"uredi conjectured bound $\mathrm{q}(\mathcal{H})\le \Delta([\mathcal{H}]_2)+1$ under certain hypotheses on the antirank $\mathrm{ar}(\mathcal{H})$ or on the…
This paper is concerned with two conjectures which are intimately related. The first is a generalization to hypergraphs of Vizing's Theorem on the chromatic index of a graph and the second is the well-known conjecture of Erd\H{o}s, Faber…
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…
In 1973 P. Erd\H{o}s and L. Lov\'asz 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, Erd\H{o}s and Lov\'asz proved that…
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…
Hoffman proved that for a simple graph $G$, the chromatic number $\chi(G)$ obeys $\chi(G) \le 1 - \frac{\lambda_1}{\lambda_{n}}$ where $\lambda_1$ and $\lambda_n$ are the maximal and minimal eigenvalues of the adjacency matrix of $G$…
Given a graph $H$, let us denote by $f_\chi(H)$ and $f_\ell(H)$, respectively, the maximum chromatic number and the maximum list chromatic number of $H$-minor-free graphs. Hadwiger's famous coloring conjecture from 1943 states that…
Erd\H{o}s and Lov\'asz noticed that an $r$-uniform intersecting hypergraph $H$ with maximal covering number, that is $\tau(H)=r$, must have at least $\frac{8}{3}r-3$ edges. There has been no improvement on this lower bound for 45 years. We…
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…
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, Erd\H{o}s and Lov\'asz asked:…
The long-standing Erd\H{o}s-Faber-Lov\'asz 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 famous conjecture of Erd\H{o}s asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem…
Let $ H = (V,E) $ be a hypergraph. By the chromatic number of a hypergraph $ H = (V,E) $ we mean the minimum number $\chi(H)$ of colors needed to paint all the vertices in $ V $ so that any edge $ e \in E $ contains at least two vertices of…
In the way of proving Kneser's conjecture, L\'{a}szl\'{o} Lov\'{a}sz settled out a new lower bound for the chromatic number of graphs. He showed that if the hom complex $||Hom(\mathcal{K}_2, H)||$ of a graph $H$ is topologically…