Perfect graphs of arbitrarily large clique-chromatic number

TL;DR

This paper demonstrates that perfect graphs can have arbitrarily large clique-chromatic numbers, constructed through a specific gluing process, answering a longstanding open question.

Contribution

It introduces a method to construct perfect graphs with large clique-chromatic numbers, expanding understanding of graph coloring properties.

Findings

Existence of perfect graphs with arbitrarily large clique-chromatic number

Construction method via gluing along cliques from cobipartite graphs

Negative answer to a previously posed question

Abstract

We prove that there exist perfect graphs of arbitrarily large clique-chromatic number. These graphs can be obtained from cobipartite graphs by repeatedly gluing along cliques. This negatively answers a question raised by Duffus, Sands, Sauer, and Woodrow in [Two-coloring all two-element maximal antichains, J. Combinatorial Theory, Ser. A, 57 (1991), 109-116].