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arXiv · 1011.4888

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

Juan José Montellano-Ballesteros·Eduardo Rivera-Campo
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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.

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Juan José Montellano-Ballesteros, Eduardo Rivera-Campo. 2010-11-22. On the heterochromatic number of hypergraphs associated to geometric graphs and to matroids. https://doi.org/10.1007/s00373-012-1190-y

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