A note on the Erdos-Faber-Lovasz Conjecture: quasigroups and complete digraphs

A decomposition of a simple graph G is a pair (G, P) where P is a set of subgraphs of G, which partitions the edges of G in the sense that every edge of G belongs to exactly one subgraph in P. If the elements of P are induced subgraphs then the decomposition is denoted by [G, P]. A k-P- coloring of a decomposition (G, P) is a surjective function that assigns to the edges of G a color from a k-set of colors, such that all edges of H is an element of P have the same color, and, if H-1, H-2 is an element of P with V(H-1)boolean AND V(H-2) not equal 0 then E(H-1) and E(H-2) have different colors. The chromatic index x' ((G, P)) of a decomposition (G, P) is the smallest number k for which there exists a k-P-coloring of (G, P). The well-known Erdos-Faber-Lovasz Conjecture states that any decomposition [K-n, P] satisfies x' ([K-n, P]) <= n. We use quasigroups and complete digraphs to give a new family of decompositions that satisfy the conjecture.