On Erdos-Faber-Lovasz Conjecture

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 same color are in the same edge. In 1978, Deza, Erd\"{o}s 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 $G$ is $n$. The clique degree $d^K(v)$ of a vertex $v$ in $G$ is given by $d^K(v) = |\{A_i: v \in V(A_i), 1 \leq i \leq n\}|$. In this paper we give a method for assigning colors to the graphs satisfying the hypothesis of the Erd\"os - Faber - Lov\'asz conjecture using intersection matrix of the cliques $A_i$'s of $G$ and clique degrees of the vertices of $G$. Also, we give theoretical proof of the conjecture for some class of graphs. In particular we show that: 1. If $G$ is a graph satisfying the hypothesis of the Conjecture 1.2 and every $A_i$ ($1 \leq i \leq n$) has at most $\sqrt{n}$ vertices of clique degree greater than 1, then $G$ is $n$-colorable. 2. If $G$ is a graph satisfying the hypothesis of the Conjecture 1.2 and every $A_i$ ($1 \leq i \leq n$) has at most $\left \lceil {\frac{n+d-1}{d}} \right \rceil$ vertices of clique degree greater than or equal to $d$ ($2\leq d \leq n$), then $G$ is $n$-colorable.