Maximal ambiguously k-colorable graphs

TL;DR

This paper characterizes maximally ambiguously k-colorable graphs using quadratic matrices, determines the maximum number of edges such graphs can have, and describes the extremal graphs.

Contribution

It provides a complete characterization of maximally ambiguously k-colorable graphs and their extremal properties.

Findings

Characterization of maximally ambiguously k-colorable graphs

Maximum number of edges in such graphs

Description of extremal 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 maximally ambiguously k-colorable graphs in terms of quadratic matrices. As an application, we calculate the maximum number of edges an ambiguously k-colorable graph can have, and characterize the extremal graphs.