Chromatic Coefficients of Linear Uniform Hypergraphs

A simple hypergraph H=(X, E), with order |X| and size m=|E|, consists of a vertex-set V(H)=X and an edge-set E(H)=E, where E X and |E| 2 for each edge E in E. H is linear if no two edges intersect in more than one vertex, and H is h-uniform, or is an h-hypergraph, if |E|=h for each E in E. The number of edges containing a vertex x is its degree dH(x). Two vertices u, v of H are in the same component if there are vertices x0=u, x1 , ..., xk=v and edges E1 , ..., Ek of H such that xi&1 , xi # Ei for each i (1 i k). If H has only one component then it is connected. A cycle C of length k in H [1] is a subhypergraph comprising k distinct vertices x1 , ..., xk and k distinct edges E1 , ..., Ek of H such that xi&1 , xi # Ei for each i (1 i k, indices taken modulo k). C is elementary if dC(xi)=2 for each i and dC( y)=1 for each other vertex y in i=1 Ei . We shall denote an elementary h-uniform cycle with m edges by C m ; clearly it has order m(h&1). An elementary path can be defined in a similar way. An h-uniform hypertree is a connected linear h-hypergraph without cycles.