4,478 research outputs found
Distinguishing Number for some Circulant Graphs
Introduced by Albertson et al. \cite{albertson}, the distinguishing number
of a graph is the least integer such that there is a
-labeling of the vertices of that is not preserved by any nontrivial
automorphism of . Most of graphs studied in literature have 2 as a
distinguishing number value except complete, multipartite graphs or cartesian
product of complete graphs depending on . In this paper, we study circulant
graphs of order where the adjacency is defined using a symmetric subset
of , called generator. We give a construction of a family of
circulant graphs of order and we show that this class has distinct
distinguishing numbers and these lasters are not depending on
Boxicity and Cubicity of Product Graphs
The 'boxicity' ('cubicity') of a graph G is the minimum natural number k such
that G can be represented as an intersection graph of axis-parallel rectangular
boxes (axis-parallel unit cubes) in . In this article, we give estimates
on the boxicity and the cubicity of Cartesian, strong and direct products of
graphs in terms of invariants of the component graphs. In particular, we study
the growth, as a function of , of the boxicity and the cubicity of the
-th power of a graph with respect to the three products. Among others, we
show a surprising result that the boxicity and the cubicity of the -th
Cartesian power of any given finite graph is in and
, respectively. On the other hand, we show that there
cannot exist any sublinear bound on the growth of the boxicity of powers of a
general graph with respect to strong and direct products.Comment: 14 page
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