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.
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Montellano-Ballesteros, J.J., Rivera-Campo, E. On the Heterochromatic Number of Hypergraphs Associated to Geometric Graphs and to Matroids. Graphs and Combinatorics 29, 1517–1522 (2013). https://doi.org/10.1007/s00373-012-1190-y
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DOI: https://doi.org/10.1007/s00373-012-1190-y