Circular-imperfection of triangle-free graphs

Circular-perfect graphs form a natural superclass of the well-known perfect graphs by means of a more general coloring concept.For perfect graphs, a characterization by means of forbidden subgraphs was recently settled by Chudnovsky et al. [Chudnovsky, M., N. Robertson, P. Seymour, and R. Thomas, The Strong Perfect Graph Theorem, Annals of Mathematics 164 (2006) 51–229]. It is, therefore, natural to ask for an analogous characterization for circular-perfect graphs or, equivalently, for a characterization of all minimally circular-imperfect graphs.Our focus is the circular-(im)perfection of triangle-free graphs. We exhibit several different new infinite families of minimally circular-imperfect triangle-free graphs. This shows that a characterization of circular-perfect graphs by means of forbidden subgraphs is a difficult task, even if restricted to the class of triangle-free graphs. This is in contrary to the perfect case where it is long-time known that the only minimally imperfect triangle-free graphs are the odd holes [Tucker, A., Critical Perfect Graphs and Perfect 3-chromatic Graphs, J. Combin. Theory (B) 23 (1977) 143–149].