Set colourings of graphs


An r-set colouring of a graph G is an assignment of r distinct colours to each vertex of G so that the sets of colours assigned to adjacent vertices are disjoint. We denote by χ(r) (G) the minimum number of colours required to r-set colour G. The set-chromatic number of G, denoted by χ* (G), is defined byχ *(G) = under(inf, r) frac(χ(r) (G), r) . Clearly 2 ≤χ* (G) ≤ χ (G). By making use of a recent result of L. Lovász we prove that min { χ(r) (G) : χ(G) = k } = 2 r + k - 2 and min { χ(r) (G) : G is uniquely k - colourable } = 2 r + k - 1. In particular, given any k, χ* (G) may be arbitrarily close to 2, and given a ny r, the ratio χ(r) (H) / χ (H) may be arbitrarily close to 1, even if H is uniquely colourable. The results disprove a conjecture of D.P. Geller. © 1979.

Publication Title

Discrete Mathematics