Further results on Erdős–Faber–Lovász conjecture

Abstract

In 1972, Erdős–Faber–Lovász (EFL) conjectured that, if 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=⋃i=1nAi denote a graph with n complete graphs A1,A2,…,An, 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 G is n. The clique degree dK(v) of a vertex v in G is given by dK(v)=|{Ai:v∈V(Ai),1≤i≤n}|. In this paper we give a method for assigning colors to the graphs satisfying the hypothesis of the Erdős–Faber–Lovász conjecture and every Ai (1≤i≤n) has atmost n2 vertices of clique degree greater than one using Symmetric latin Squares and clique degrees of the vertices of G.