On the Heterochromatic Number of Hypergraphs Associated to Geometric Graphs and to Matroids

Abstract

The heterochromatic number h c (H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a hyperedge of H all of whose vertices have different colours. We denote by ν(H) the number of vertices of H and by τ(H) the size of the smallest set containing at least two vertices of each hyperedge of H. For a complete geometric graph G with n ≥ 3 vertices let H = H(G) be the hypergraph whose vertices are the edges of G and whose hyperedges are the edge sets of plane spanning trees of G. We prove that if G has at most one interior vertex, then h c (H) = ν(H) − τ(H) + 2. We also show that h c (H) = ν(H) − τ(H) + 2 whenever H is a hypergraph with vertex set and hyperedge set given by the ground set and the bases of a matroid, respectively.