257 research outputs found
Stability number and f-factors in graphs
We present a new sufficient condition on stability number and toughness of
the graph to have an f-factor
Vertex-Coloring 2-Edge-Weighting of Graphs
A -{\it edge-weighting} of a graph is an assignment of an integer
weight, , to each edge . An edge weighting naturally
induces a vertex coloring by defining for every
. A -edge-weighting of a graph is \emph{vertex-coloring} if
the induced coloring is proper, i.e., for any edge .
Given a graph and a vertex coloring , does there exist an
edge-weighting such that the induced vertex coloring is ? We investigate
this problem by considering edge-weightings defined on an abelian group.
It was proved that every 3-colorable graph admits a vertex-coloring
-edge-weighting \cite{KLT}. Does every 2-colorable graph (i.e., bipartite
graphs) admit a vertex-coloring 2-edge-weighting? We obtain several simple
sufficient conditions for graphs to be vertex-coloring 2-edge-weighting. In
particular, we show that 3-connected bipartite graphs admit vertex-coloring
2-edge-weighting
Degree Sequences and the Existence of -Factors
We consider sufficient conditions for a degree sequence to be forcibly
-factor graphical. We note that previous work on degrees and factors has
focused primarily on finding conditions for a degree sequence to be potentially
-factor graphical.
We first give a theorem for to be forcibly 1-factor graphical and, more
generally, forcibly graphical with deficiency at most . These
theorems are equal in strength to Chv\'atal's well-known hamiltonian theorem,
i.e., the best monotone degree condition for hamiltonicity. We then give an
equally strong theorem for to be forcibly 2-factor graphical.
Unfortunately, the number of nonredundant conditions that must be checked
increases significantly in moving from to , and we conjecture that
the number of nonredundant conditions in a best monotone theorem for a
-factor will increase superpolynomially in .
This suggests the desirability of finding a theorem for to be forcibly
-factor graphical whose algorithmic complexity grows more slowly. In the
final section, we present such a theorem for any , based on Tutte's
well-known factor theorem. While this theorem is not best monotone, we show
that it is nevertheless tight in a precise way, and give examples illustrating
this tightness.Comment: 19 page
Sampling and Reconstruction of Signals on Product Graphs
In this paper, we consider the problem of subsampling and reconstruction of
signals that reside on the vertices of a product graph, such as sensor network
time series, genomic signals, or product ratings in a social network.
Specifically, we leverage the product structure of the underlying domain and
sample nodes from the graph factors. The proposed scheme is particularly useful
for processing signals on large-scale product graphs. The sampling sets are
designed using a low-complexity greedy algorithm and can be proven to be
near-optimal. To illustrate the developed theory, numerical experiments based
on real datasets are provided for sampling 3D dynamic point clouds and for
active learning in recommender systems.Comment: 5 pages, 3 figure
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