Clique partitions and distinct color sets in list colorings

Abstract

For an \(n\)-vertex graph \(G\) and an integer \(m\ge n\), let \(f'(G,m)\) be the minimum number of distinct color sets used by proper colorings from \(m\)-element lists that are disjoint at distinct nonadjacent vertices. A clique-partition assignment gives a common list to the vertices of each part of a clique partition and uses disjoint lists on different parts. In this paper, we construct a graph on \(20\) vertices for which these assignments do not attain \(f'(G,20)\), and a fixed graph on \(25\) vertices for which they fail to attain the minimum for every allowed list size. For every triangle-free graph \(G\) on \(n\ge2\) vertices, we prove \[ f'(G,m)=\binom{m}{2}^{ν(G)}m^{n-2ν(G)} \qquad (m\ge n), \] where \(ν(G)\) is the size of a maximum matching, and characterize all minimizing assignments in terms of maximum matchings. For clique blow-ups of each fixed odd cycle of length at least five, we determine the minimum up to constant factors uniformly over the blow-up size and all allowed list sizes, and obtain an exponential separation from the clique-partition value. We also prove that a noncomplete graph \(G\) satisfies \(f'(G,n)=n^2\) if and only if all edges of its complement have a common endpoint. These results disprove the general clique-partition conjecture of Gravier, Maffray, and Mohar (\emph{Discrete Mathematics}, 2003) and its extension by Wu and Zhang (\emph{Applied Mathematics Letters}, 2006), while establishing both triangle-free conjectures.