Maximal ambiguously k-colorable graphs

Abstract

A graph is ambiguously k-colorable if its vertex set admits two distinct partitions each into at most k anticliques. We give a full characterization of the maximal ambiguously k-colorable graphs in terms of $k\times k$-matrices. As an application, we calculate the maximum number of edges an ambiguously k-colorable graph can have, and characterize the extremal graphs.