Characterization of extremal graphs

Characterize the graphs attaining the maximum ordinary villainy M(n, k) and the maximum weak villainy Mw(n, k) among all n-vertex graphs of chromatic number k, and determine which edges can be removed from a complete multipartite extremizer while preserving both its chromatic number and its extremal villainy.

Background

The paper determines the extremal values M(n,k) and Mw(n,k) and proves that complete multipartite graphs provide extremizers. However, attaining the same extremal value need not force a graph itself to be complete multipartite.

The open problem asks for a structural classification of all extremal graphs and, more specifically, for a description of edge deletions from complete multipartite extremizers that preserve chromatic number and the relevant extremal villainy.