# On 2-periodic graphs of a certain graph operator

• Volume: 21, Issue: 1, page 13-30
• ISSN: 2083-5892

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## Abstract

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We deal with the graph operator $\overline{Pow₂}$ defined to be the complement of the square of a graph: $\overline{Pow₂}\left(G\right)=\overline{Pow₂\left(G\right)}$. 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 belong to and find infinitely many graphs also belonging to among generalized hypercubes.

## How to cite

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Ivan Havel, and Bohdan Zelinka. "On 2-periodic graphs of a certain graph operator." Discussiones Mathematicae Graph Theory 21.1 (2001): 13-30. <http://eudml.org/doc/270637>.

@article{IvanHavel2001,
abstract = {We deal with the graph operator $\overline\{Pow₂\}$ defined to be the complement of the square of a graph: $\overline\{Pow₂\}(G) = \overline\{Pow₂(G)\}$. 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 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 belong to and find infinitely many graphs also belonging to among generalized hypercubes.},
author = {Ivan Havel, Bohdan Zelinka},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph operator; power and complement of a graph; Desarguesian finite projective geometry; decomposition of a complete bipartite graph; generalized hypercube; power of graphs; generalized hypercubes; decomposition of complete bipartite graphs},
language = {eng},
number = {1},
pages = {13-30},
title = {On 2-periodic graphs of a certain graph operator},
url = {http://eudml.org/doc/270637},
volume = {21},
year = {2001},
}

TY - JOUR
AU - Ivan Havel
TI - On 2-periodic graphs of a certain graph operator
JO - Discussiones Mathematicae Graph Theory
PY - 2001
VL - 21
IS - 1
SP - 13
EP - 30
AB - We deal with the graph operator $\overline{Pow₂}$ defined to be the complement of the square of a graph: $\overline{Pow₂}(G) = \overline{Pow₂(G)}$. 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 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 belong to and find infinitely many graphs also belonging to among generalized hypercubes.
LA - eng
KW - graph operator; power and complement of a graph; Desarguesian finite projective geometry; decomposition of a complete bipartite graph; generalized hypercube; power of graphs; generalized hypercubes; decomposition of complete bipartite graphs
UR - http://eudml.org/doc/270637
ER -

## References

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1. [1] J. Akiyama, H. Era and G. Exoo, Further results on graph equations for line graphs and n'th power graphs, Discrete Math. 34 (1981) 209-218, doi: 10.1016/0012-365X(81)90001-7. Zbl0464.05057
2. [2] T. Dvorák, I. Havel, J-M. Laborde and P. Liebl, Generalized hypercubes and graph embedding with dilation, Rostocker Mathematisches Kolloquium 39 (1990) 13-20. Zbl0719.05036
3. [3] M. Hall, Jr., Combinatorial Theory (Blaisdell Publishing Company, Waltham (Massachusetts) - Toronto - London 1967).
4. [4] F. Harary, Four difficult unsolved problems in graph theory, in: M. Fiedler ed., Recent Advances in Graph Theory, Proc. Second Czech. Symp. Graph Theory, Prague, 1974 (Academia, Praha, 1975) 249-256.
5. [5] Ch. Payan, On the chromatic number of cube-like graphs, Discrete Math. 103 (1992) 271-277, doi: 10.1016/0012-365X(92)90319-B. Zbl0772.05043
6. [6] E. Prisner, Graph dynamics (Pitman Research Notes in Mathematics Series, 338, 1995).

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