Coloring Non-uniform Hypergraphs Without Short Cycles

Abstract

The work deals with a generalization of Erd�s---Lovasz problem concerning colorings of non-uniform hypergraphs. Let H = (V, E) be a hypergraph and let $${{f_r(H)=\sum\limits_{e \in E}r^{1-|e|}}}$$ f r ( H ) = � e � E r 1 - | e | for some r � 2. Erd�s and Lovasz proposed to find the value f (n) equal to the minimum possible value of f 2(H) where H is 3-chromatic hypergraph with minimum edge-cardinality n. In the paper we study similar problem for the class of hypergraphs with large girth. We prove that if H is a hypergraph with minimum edge-cardinality n � 3 and girth at least 4, satisfying the inequality $$f_r(H) \leq \frac{1}{2}\, \left(\frac{n}{{\rm ln}\, n}\right)^{2/3},$$ f r ( H ) ≤ 1 2 n ln n 2 / 3 , then H is r -colorable. Our result improves previous lower bounds for f (n) in the class of hypergraphs without 2- and 3-cycles.