If \(G\) is \((a, b)\)-choosable, then \(G\) is \((am, bm)\)-choosable for every positive integer \(m\)

Choosability of graphs

A graph \( G\) is said to be \( (a,b)\)-choosable if for any assignment of a list of \( a\) colors to each of its vertices there is a subset of \( b\) colors of each list so that subsets corresponding to adjacent vertices are disjoint.

Conjecture (proposed by Erdös, Rubin, and Taylor):

If \(G\) is \( (a,b)\)-choosable, then \(G\) is \( (am,bm)\)-choosable for every positive integer \(m\).

1 P. Erdös, A. L. Rubin and H. Taylor, Choosability in graphs, Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing (Humboldt State Univ., Arcata, Calif., 1979), Congress. Numer. XXVI, 125-157, Utilitas Math., Winnipeg, Man., 1980.