# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2017-2018): c. 84%

# Discussiones Mathematicae Graph Theory

## ON 2-PERIODIC GRAPHS OF A CERTAIN GRAPH OPERATOR

 Ivan Havel Mathematical Institute, Academy of Sciences of the Czech Republic Zitná 25, 115 67 Prague 1, Czech Republic Bohdan Zelinka Technical University Voronezská 13, 461 17 Liberec, Czech Republic e-mail: Bohdan.Zelinka@vslib.cz

## Abstract

We deal with the graph operator [(Pow2)] defined to be the complement of the square of a graph: [(Pow2)](G) = [`(Pow2(G))]. Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class G of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph Km,n can be decomposed in two edge-disjoint factors from G. We further show that all the incidence graphs of Desarguesian finite projective geometries belong to G and find infinitely many graphs also belonging to G among generalized hypercubes.

Keywords: graph operator, power and complement of a graph, Desarguesian finite projective geometry, decomposition of a complete bipartite graph, generalized hypercube.

2000 Mathematics Subject Classification: 05C38, 05C75.

## References

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