1,072 research outputs found
Saturation Number of Trees in the Hypercube
A graph is -saturated if it is -free and the addition
of any edge of not in creates a copy of . The saturation
number is the minimum number of edges in a -saturated
graph. We investigate bounds on the saturation number of trees in the
-dimensional hypercube . We first present a general lower bound on the
saturation number based on the minimum degree of non-leaves. From there, we
suggest two general methods for constructing -saturated subgraphs of ,
and prove nontrivial upper bounds for specific types of trees, including paths,
generalized stars, and certain caterpillars under a restriction on minimum
degree with respect to diameter.Comment: 21 page
Polychromatic Colorings on the Hypercube
Given a subgraph G of the hypercube Q_n, a coloring of the edges of Q_n such
that every embedding of G contains an edge of every color is called a
G-polychromatic coloring. The maximum number of colors with which it is
possible to G-polychromatically color the edges of any hypercube is called the
polychromatic number of G. To determine polychromatic numbers, it is only
necessary to consider a structured class of colorings, which we call simple.
The main tool for finding upper bounds on polychromatic numbers is to translate
the question of polychromatically coloring the hypercube so every embedding of
a graph G contains every color into a question of coloring the 2-dimensional
grid so that every so-called shape sequence corresponding to G contains every
color. After surveying the tools for finding polychromatic numbers, we apply
these techniques to find polychromatic numbers of a class of graphs called
punctured hypercubes. We also consider the problem of finding polychromatic
numbers in the setting where larger subcubes of the hypercube are colored. We
exhibit two new constructions which show that this problem is not a
straightforward generalization of the edge coloring problem.Comment: 24 page
Embeddings into the Pancake Interconnection Network
Owing to its nice properties, the pancake is one of the Cayley graphs that
were proposed as alternatives to the hypercube for interconnecting processors
in parallel computers. In this paper, we present embeddings of rings, grids and
hypercubes into the pancake with constant dilation and congestion. We also
extend the results to similar efficient embeddings into the star graph.Comment: Article paru en 2002 dans Parallel Processing Letter
Embedded connectivity of recursive networks
Let be an -dimensional recursive network. The -embedded
connectivity (resp. edge-connectivity ) of is
the minimum number of vertices (resp. edges) whose removal results in
disconnected and each vertex is contained in an -dimensional subnetwork
. This paper determines and for the hypercube and
the star graph , and for the bubble-sort network
On the star arboricity of hypercubes
A Hypercube is a graph in which the vertices are all binary vectors of
length n, and two vertices are adjacent if and only if their components differ
in exactly one place. A galaxy or a star forest is a union of vertex disjoint
stars. The star arboricity of a graph , , is the minimum number
of galaxies which partition the edge set of . In this paper among other
results, we determine the exact values of for , . We also improve the last known
upper bound of and show the relation between and
square coloring.Comment: Australas. J. Combin., vol. 59 pt. 2, (2014
Hypercube emulation of interconnection networks topologies
We address various topologies (de Bruijn, chordal ring, generalized Petersen,
meshes) in various ways ( isometric embedding, embedding up to scale, embedding
up to a distance) in a hypercube or a half-hypercube. Example of obtained
embeddings: infinite series of hypercube embeddable Bubble Sort and Double
Chordal Rings topologies, as well as of regular maps.Comment: 14 pages, 5 tables, 7 figure
Parity Edge-Coloring of Graphs
In a graph whose edges are colored, a parity walk is a walk that uses each
color an even number of times. The parity edge chromatic number p(G) of a graph
G is the least k so that there is a coloring of E(G) using k colors that does
not contain a parity path. The strong parity edge chromatic number p'(G) of G
is the least k so that there is a coloring of E(G) using k colors with the
property that every parity walk is closed.
Our main result is to determine p'(K_n). Specifically, if m is the least
power of two that is as large as n, then p'(K_n) has value m - 1. As a
corollary, we strengthen a special case of an old result of Daykin and Lovasz.
Other results include determining p(G) and p'(G) whenever G is a path, cycle,
or of the form K_{2,n}, and an upper bound on p'(G) for the case that G is a
complete bipartite graph. We conclude with a sample of open problems.Comment: 23 pages, 0 figure
Biclique Covers and Partitions
The biclique cover number (resp. biclique partition number) of a graph ,
) (resp. ), is the least number of biclique
(complete bipartite) subgraphs that are needed to cover (resp. partition) the
edges of . The \emph{local biclique cover number} (resp. local biclique
partition number) of a graph , ) (resp. ),
is the least such that there is a cover (resp. partition) of the edges of
by bicliques with no vertex in more than of these bicliques. We show
that may be bounded in terms of , in
particular, . However, the
analogous result does not hold for the local measures. Indeed, in our main
result, we show that can be arbitrarily large, even for
graphs with . For such graphs, , we try to bound
in terms of additional information about biclique covers of
. We both answer and leave open questions related to this. There is a well
known link between biclique covers and subcube intersection graphs. We consider
the problem of finding the least for which every graph on vertices
can be represented as a subcube intersection graph in which every subcube has
dimension . We reduce this problem to the much studied question of finding
the least such that every graph on vertices is the intersection
graph of subcubes of a -dimensional cube.Comment: 12 pages, Journal copy; typos corrected, reference added, Electronic
Journal of Combinatorics, Volume 21, Issue 1, 201
Distance and routing labeling schemes for cube-free median graphs
Distance labeling schemes are schemes that label the vertices of a graph with
short labels in such a way that the distance between any two vertices and
can be determined efficiently by merely inspecting the labels of and
, without using any other information. Similarly, routing labeling schemes
label the vertices of a graph in a such a way that given the labels of a source
node and a destination node, it is possible to compute efficiently the port
number of the edge from the source that heads in the direction of the
destination. One of important problems is finding natural classes of graphs
admitting distance and/or routing labeling schemes with labels of
polylogarithmic size. In this paper, we show that the class of cube-free median
graphs on nodes enjoys distance and routing labeling schemes with labels of
bits.Comment: 34 pages, 10 figure
On simplicial and cubical complexes with short links
We consider closed simplicial and cubical -complexes in terms of link of
their -faces. Especially, we consider the case, when this link has size
3 or 4, i.e., every -face is contained in 3 or 4 -faces. Such
simplicial complexes with {\em short} (i.e. of length 3 or 4) links are
completely classified by their {\em characteristic partition}. We consider also
embedding into hypercubes of the skeletons of simplicial and cubical complexes.Comment: 10 pages, 1 table, 3 figure
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