The Normal Graph Conjecture is true for Circulants
نویسندگان
چکیده
Normal graphs are defined in terms of cross-intersecting set families: a graph is normal if it admits a clique cover Q and a stable set cover S s.t. every clique in Q intersects every stable set in S. Normal graphs can be considered as closure of perfect graphs by means of co-normal products (K ̈orner [6]) and graph entropy (Czisz ́ar et al. [5]). Perfect graphs have been recently characterized as those graphs without odd holes and odd antiholes as induced subgraphs (Strong Perfect Graph Theorem, Chudnovsky et al. [3]). K ̈orner and de Simone [9] observed that C5,C7, and C7 are minimal not normal and conjectured, as generalization of the Strong Perfect Graph Theorem, that every C5, C7, C7-free graph is normal (Normal Graph Conjecture, K ̈orner and de Simone [9]). We prove this conjecture for a first class of graphs that generalize both odd holes and odd antiholes, the circulants, by characterizing all the normal circulants.
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