Minimal normal graph covers

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Abstract

A graph is normal if it admits a clique cover C and a stable set cover S such that each clique in C and each stable set in S have a vertex in common. The pair (C,S) is a normal cover of the graph. We present the following extremal property of normal covers. For positive integers c, s, if a graph with n vertices admits a normal cover with cliques of sizes at most c and stable sets of sizes at most s, then c+s≥log2(n). For infinitely many n, we also give a construction of a graph with n vertices that admits a normal cover with cliques and stable sets of sizes less than 0.87log2(n). Furthermore, we show that for all n, there exists a normal graph with n vertices, clique number Θ(log2(n)) and independence number Θ(log2(n)).

When c or s are very small, we can describe all normal graphs with the largest possible number of vertices that allow a normal cover with cliques of sizes at most c and stable sets of sizes at most s. However, such extremal graphs remain elusive even for moderately small values of c and s.

Mathematics Subject Classification (2000)

05C35 05C69 05C70 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.FMF, Department of MathematicsUniversity of LjubljanaLjubljanaSlovenia
  2. 2.Department of MathematicsSimon Fraser UniversityBurnabyCanada
  3. 3.On leave from: IMFM & FMF, Department of MathematicsUniversity of LjubljanaLjubljanaSlovenia

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