We consider the Erd{H{o}}s - Faber - Lov{a}sz (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 $r ge 4$, $chi (textbf{H}) le 1.181n$ and if $r=3$, $chi(textbf{H}) le 1.281n$
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