3 research outputs found
The Z-cubes: a hypercube variant with small diameter
This paper introduces a new variant of hypercubes, which we call Z-cubes. The
n-dimensional Z-cube is obtained from two copies of the (n-1)-dimensional
Z-cube by adding a special perfect matching between the vertices of
these two copies of . We prove that the n-dimensional Z-cubes
has diameter . This greatly improves on the previous known
variants of hypercube of dimension n, whose diameters are all larger than n/3.
Moreover, any hypercube variant of dimension is an n-regular graph on
vertices, and hence has diameter greater than . So the Z-cubes are
optimal with respect to diameters, up to an error of order .
Another type of Z-cubes which have similar structure and properties
as are also discussed in the last section.Comment: 9 pages, 1 figur
A note on minimum linear arrangement for BC graphs
A linear arrangement is a labeling or a numbering or a linear ordering of the
vertices of a graph. In this paper we solve the minimum linear arrangement
problem for bijective connection graphs (for short BC graphs) which include
hypercubes, M\"{o}bius cubes, crossed cubes, twisted cubes, locally twisted
cube, spined cube, -cubes, etc. as the subfamilies.Comment: 6 page
Fractional matching preclusion of fault Hamiltonian graphs
Matching preclusion is a measure of robustness in the event of edge failure
in interconnection networks. As a generalization of matching preclusion, the
fractional matching preclusion number (FMP number for short) of a graph is the
minimum number of edges whose deletion results in a graph that has no
fractional perfect matchings, and the fractional strong matching preclusion
number (FSMP number for short) of a graph is the minimum number of edges and/or
vertices whose deletion leaves a resulting graph with no fractional perfect
matchings. A graph is said to be -fault Hamiltonian if there exists a
Hamiltonian cycle in for any set of vertices and/or edges with
. In this paper, we establish the FMP number and FSMP number of
-fault Hamiltonian graphs with minimum degree . As
applications, the FMP number and FSMP number of some well-known networks are
determined