Combinatorica

pp 1–22

# Minimal normal graph covers

Article

## 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

## References

1. [1]
I. Csiszár and J. Körner: Information theory, Cambridge University Press, second edition, 2011.
2. [2]
I. Csiszár, J. Körner, L. Lovász, K. Marton and G. Simonyi: Entropy splitting for antiblocking corners and perfect graphs, Combinatorica 10 (1990), 27–40.
3. [3]
C. De Simone and J. Körner: On the odd cycles of normal graphs, Discrete Appl. Math. 94 (1999), 161–169.
4. [4]
E. Fachini and J. Körner: Cross-intersecting couples of graphs, J. Graph Theory 56 (2007), 105–112.
5. [5]
A. Harutyunyan, L. Pastor and S. Thomasse: Disproving the normal graph conjecture, 2015, Preprint, arXiv:1508.05487.Google Scholar
6. [6]
J. Körner: An extension of the class of perfect graphs, Studia Sci. Math. Hungar. 8 (1973), 405–409.
7. [7]
J. Körner and G. Longo: Two-step encoding for finite sources, IEEE Trans. Inf. Theor. 19 (2006), 778–782.
8. [8]
J. Körner, G. Simonyi and Z. Tuza: Perfect couples of graphs, Combinatorica 12 (1992), 179–192.
9. [9]
L. Lovász: Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972), 253–267.
10. [10]
I. E. Zverovich and V. E. Zverovich: Basic perfect graphs and their extensions, Discrete Math. 293 (2005), 291–311.

© 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