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On 2-periodic graphs of a certain graph operator

Ivan Havel, Bohdan Zelinka (2001)

Discussiones Mathematicae Graph Theory

We deal with the graph operator P o w ₂ ¯ defined to be the complement of the square of a graph: P o w ₂ ¯ ( G ) = P o w ₂ ( 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 of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph K m , n can be decomposed in two edge-disjoint factors from . We further show that all the incidence graphs of Desarguesian finite projective geometries...

On a problem of walks

Charles Delorme, Marie-Claude Heydemann (1999)

Annales de l'institut Fourier

In 1995, F. Jaeger and M.-C. Heydemann began to work on a conjecture on binary operations which are related to homomorphisms of De Bruijn digraphs. For this, they have considered the class of digraphs G such that for any integer k , G has exactly n walks of length k , where n is the order of G . Recently, C. Delorme has obtained some results on the original conjecture. The aim of this paper is to recall the conjecture and to report where all the authors arrived.

On arbitrarily vertex decomposable unicyclic graphs with dominating cycle

Sylwia Cichacz, Irmina A. Zioło (2006)

Discussiones Mathematicae Graph Theory

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n₁,...,nₖ) of positive integers such that ∑ i = 1 k n i = n , there exists a partition (V₁,...,Vₖ) of vertex set of G such that for every i ∈ 1,...,k the set V i induces a connected subgraph of G on n i vertices. We consider arbitrarily vertex decomposable unicyclic graphs with dominating cycle. We also characterize all such graphs with at most four hanging vertices such that exactly two of them have a common neighbour.

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