The problem of embedding graphs into other graphs is much studied in the graph theory. In fact, much effort has been devoted to determining the conditions under which a graph G is a subgraph of a graph H, having a particular structure. An important class to study is the set of graphs which are embeddable into a hypercube. This importance results from the remarkable properties of the hypercube and its use in several domains, such as: the coding theory, transfer of information, multicriteria rule,...
A -graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of -graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free -graph. -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 -graphs and more specifically in -graphs,...
The problem of embedding graphs into other graphs is much studied in the
graph theory. In fact, much effort has been devoted to determining the
conditions under which a graph G is a subgraph of a graph H, having a
particular structure. An important class to study is the set of graphs which
are embeddable into a hypercube. This importance results from the remarkable
properties of the hypercube and its use in several domains, such as: the
coding theory, transfer of information, multicriteria rule,...
The cubical dimension of a graph is the smallest dimension of a hypercube into which is embeddable as a subgraph. The conjecture of Havel (1984) claims that the cubical dimension of every balanced binary tree with vertices, , is . The 2-rooted complete binary tree of depth is obtained from two copies of the complete binary tree of depth by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice a 2-rooted...
The main subject of our study are spherical (weakly spherical) graphs, i.e. connected graphs fulfilling the condition that in each interval to each vertex there is exactly one (at least one, respectively) antipodal vertex. Our analysis concerns properties of these graphs especially in connection with convexity and also with hypercube graphs. We deal e.g. with the problem under what conditions all intervals of a spherical graph induce hypercubes and find a new characterization of hypercubes: is...
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