Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 29-36, 2017

# 4-cycle properties for characterizing rectagraphs and hypercubes

#### Received May 1, 2015.   First published February 24, 2017.

Khadra Bouanane, Department of Mathematics, Kasdi Merbah University, Ave 1er Novembre 1954, 30000 Ouargla, Algeria, and Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El Alia, 16111 Bab Ezzouar, Algiers, Algeria, e-mail: bouanane.khadra@univ-ouargla.dz; Abdelhafid Berrachedi, Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El Alia, 16111 Bab Ezzouar, Algiers, Algeria, e-mail: berrachedi.abdelhafid@usthb.dz

Abstract: A $(0,2)$-graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of $(0,\lambda)$-graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free $(0,2)$-graph. $(0,2)$-graphs include hypercubes, folded cube graphs and some particular graphs such as icosahedral graph, Shrikhande graph, Klein graph, Gewirtz graph, etc. In this paper, we give some local properties of 4-cycles in $(0,\lambda)$-graphs and more specifically in $(0,2)$-graphs, leading to new characterizations of rectagraphs and hypercubes.

Keywords: hypercube; $(0,2)$-graph; rectagraph; 4-cycle; characterization

Classification (MSC 2010): 05C75

DOI: 10.21136/CMJ.2017.0227-15

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