626 research outputs found
Generalized Matching Preclusion in Bipartite Graphs
The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion results in a graph that has no perfect matchings. For many interconnection networks, the optimal such sets are precisely sets of edges incident to a single vertex. The conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond these, and it is defined as the minimum number of edges whose deletion results in a graph with neither isolated vertices nor perfect matchings. In this paper we generalize this concept to get a hierarchy of stronger matching preclusion properties in bipartite graphs, and completely characterize such properties of complete bipartite graphs and hypercubes
Tight upper bound on the maximum anti-forcing numbers of graphs
Let be a simple graph with a perfect matching. Deng and Zhang showed that
the maximum anti-forcing number of is no more than the cyclomatic number.
In this paper, we get a novel upper bound on the maximum anti-forcing number of
and investigate the extremal graphs. If has a perfect matching
whose anti-forcing number attains this upper bound, then we say is an
extremal graph and is a nice perfect matching. We obtain an equivalent
condition for the nice perfect matchings of and establish a one-to-one
correspondence between the nice perfect matchings and the edge-involutions of
, which are the automorphisms of order two such that and
are adjacent for every vertex . We demonstrate that all extremal
graphs can be constructed from by implementing two expansion operations,
and is extremal if and only if one factor in a Cartesian decomposition of
is extremal. As examples, we have that all perfect matchings of the
complete graph and the complete bipartite graph are nice.
Also we show that the hypercube , the folded hypercube ()
and the enhanced hypercube () have exactly ,
and nice perfect matchings respectively.Comment: 15 pages, 7 figure
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
Parity balance of the -th dimension edges in Hamiltonian cycles of the hypercube
Let be an integer, and let . An -th dimension
edge in the -dimensional hypercube is an edge such that
differ just at their -th entries. The parity of an -th
dimension edge \edg{v_1}{v_2} is the number of 1's modulus 2 of any of its
vertex ignoring the -th entry. We prove that the number of -th dimension
edges appearing in a given Hamiltonian cycle of with parity zero
coincides with the number of edges with parity one. As an application of this
result it is introduced and explored the conjecture of the inscribed squares in
Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in contains
two opposite edges in a 4-cycle. We prove this conjecture for , and
for any Hamiltonian cycle containing more than edges in the same
dimension. This bound is finally improved considering the equi-independence
number of , which is a concept introduced in this paper for bipartite
graphs
On realization graphs of degree sequences
Given the degree sequence of a graph, the realization graph of is the
graph having as its vertices the labeled realizations of , with two vertices
adjacent if one realization may be obtained from the other via an
edge-switching operation. We describe a connection between Cartesian products
in realization graphs and the canonical decomposition of degree sequences
described by R.I. Tyshkevich and others. As applications, we characterize the
degree sequences whose realization graphs are triangle-free graphs or
hypercubes.Comment: 10 pages, 5 figure
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