2,763 research outputs found
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
Vertex decomposable graphs, codismantlability, Cohen-Macaulayness and Castelnuovo-Mumford regularity
We call a (simple) graph G codismantlable if either it has no edges or else
it has a codominated vertex x, meaning that the closed neighborhood of x
contains that of one of its neighbor, such that G-x codismantlable. We prove
that if G is well-covered and it lacks induced cycles of length four, five and
seven, than the vertex decomposability, codismantlability and
Cohen-Macaulayness for G are all equivalent. The rest deals with the
computation of Castelnuovo-Mumford regularity of codismantlable graphs. Note
that our approach complements and unifies many of the earlier results on
bipartite, chordal and very well-covered graphs
Locating-dominating sets in twin-free graphs
A locating-dominating set of a graph is a dominating set of with
the additional property that every two distinct vertices outside have
distinct neighbors in ; that is, for distinct vertices and outside
, where denotes the open neighborhood
of . A graph is twin-free if every two distinct vertices have distinct open
and closed neighborhoods. The location-domination number of , denoted
, is the minimum cardinality of a locating-dominating set in .
It is conjectured [D. Garijo, A. Gonz\'alez and A. M\'arquez. The difference
between the metric dimension and the determining number of a graph. Applied
Mathematics and Computation 249 (2014), 487--501] that if is a twin-free
graph of order without isolated vertices, then . We prove the general bound ,
slightly improving over the bound of Garijo et
al. We then provide constructions of graphs reaching the bound,
showing that if the conjecture is true, the family of extremal graphs is a very
rich one. Moreover, we characterize the trees that are extremal for this
bound. We finally prove the conjecture for split graphs and co-bipartite
graphs.Comment: 11 pages; 4 figure
Some recent results in the analysis of greedy algorithms for assignment problems
We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems
Independence and matching numbers of some token graphs
Let be a graph of order and let . The -token
graph of , is the graph whose vertices are the -subsets of
, where two vertices are adjacent in whenever their symmetric
difference is an edge of . We study the independence and matching numbers of
. We present a tight lower bound for the matching number of
for the case in which has either a perfect matching or an almost perfect
matching. Also, we estimate the independence number for bipartite -token
graphs, and determine the exact value for some graphs.Comment: 16 pages, 4 figures. Third version is a major revision. Some proofs
were corrected or simplified. New references adde
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