Coloring linear hypergraphs: the Erdős–Faber–Lovász conjecture and the Combinatorial Nullstellensatz

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, which verifies a necessary condition for our algebraic approach to work.

The Erdős–Faber–Lovász conjecture for the class of δEFL graphs

In this paper, we present the class of [Formula: see text] graphs and then prove a generalized statement of the Erdős–Faber–Lovász conjecture for this graph class. Then, the Erdős–Faber–Lovász conjecture follows for every graph in this class.

A survey on Hamiltonicity in Cayley graphs and digraphs on different groups

Lovász had posed a question stating whether every connected, vertex-transitive graph has a Hamilton path in 1969. There is a growing interest in solving this longstanding problem and still it remains widely open. In fact, it was known that only five vertex-transitive graphs exist without a Hamiltonian cycle which do not belong to Cayley graphs. A Cayley graph is the subclass of vertex-transitive graph, and in view of the Lovász conjecture, the attention has focused more toward the Hamiltonicity of Cayley graphs. This survey will describe the current status of the search for Hamiltonian cycles and paths in Cayley graphs and digraphs on different groups, and discuss the future direction regarding famous conjecture.

The Erdős–Faber–Lovász conjecture is true for n ≤ 12

The unsolved Erdős–Faber–Lovász conjecture states that if a hypergraph has n edges, each of size n, and every pair of edges intersect in at most one vertex, then its vertex chromatic number is equal to n. Hindman proved it for n ≤ 10; here we extend this result up to n ≤ 12.