Elementary Bipartite Graphs and Unique Colourability

Abstract

AbstractThe Clique-Pair-Conjecture (CPC) states that a uniquely colourable perfect graph, different from a clique, contains two maximum size cliques having a two element symmetric difference. One can make an auxiliary graphB from a minimal counterexample for the CPC (if any exists), this B is bipartite. We prove thatB is elementary. Furthermore, we give a characterization of the minimal counterexamples, in terms of the constructed auxiliary graphs