The Multiset Chromatic Number of a Graph

A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χm(G) of G. For every graph G, χm(G) is bounded above by its chromatic number χ(G). The multiset chromatic number is determined for every complete multipartite graph as well as for cycles and their squares, cubes, and fourth powers. It is conjectured that for each k > 3, there exist sufficiently large integers n such that χm(C k n) = 3. It is determined for which pairs k, n of integers with 1 6 k 6 n and n > 3, there exists a connected graph G of order n with χm(G) = k. For k = n− 2, all such graphs G are determined.