On the heterochromatic number of hypergraphs associated to geometric graphs and to matroids

TL;DR

This paper investigates the heterochromatic number of hypergraphs derived from geometric graphs and matroids, establishing exact formulas under specific conditions and expanding understanding of hypergraph coloring properties.

Contribution

It provides a formula for the heterochromatic number of hypergraphs from geometric graphs with at most one interior vertex and extends this to hypergraphs from matroids.

Findings

hc(H) = nu(H) - tau(H) + 2 for geometric graphs with at most one interior vertex

hc(H) = nu(H) - tau(H) + 2 for hypergraphs from matroids

Establishes conditions under which the heterochromatic number can be exactly determined.

Abstract

The heterochromatic number hc(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 nu(H) the number of vertices of H and by tau(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 > 2 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 hc(H) = nu(H) - tau(H) + 2. We also show that hc(H) = nu(H) - tau(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.