The Normal Graph Conjecture is True for Circulants

Abstract

Normal graphs are defined in terms of cross-intersecting set families and turned out to be “weaker” perfect graphs w.r.t. several aspects, e.g., by means of co-normal products (Korner [5]) and graph entropy (Cziszar et al. [4]). Perfect graphs have been recently characterized as those graphs without odd holes and odd antiholes as induced subgraphs (Chudnovsky et al. [2]). In analogy, Korner and de Simone [7] conjectured that every (C 5,C 7,\( \overline C _7 \))-free graph is normal (Normal Graph Conjecture). We prove this conjecture for a first class of graphs that generalize both odd holes and odd antiholes: the circulants. For that, we characterize all the normal circulants by explicitly constructing the required set families for all normal circulants and showing that the remaining ones are not (C 5,C 7,\( \overline C _7 \))-free.