292 research outputs found
Anti-Ramsey number of edge-disjoint rainbow spanning trees
An edge-colored graph is called rainbow if every edge of receives a
different color. The anti-Ramsey number of edge-disjoint rainbow spanning
trees, denoted by , is defined as the maximum number of colors in an
edge-coloring of containing no edge-disjoint rainbow spanning trees.
Jahanbekam and West [J. Graph Theory, 2014] conjectured that for any fixed ,
whenever . In this paper, we prove
this conjecture. We also determine when . Together with
previous results, this gives the anti-Ramsey number of edge-disjoint
rainbow spanning trees for all values of and .Comment: 17 pages, fixed an error in the proof of Theorem 3 using Matroid
method
A note on 1-guardable graphs in the cops and robber game
In the cops and robber games played on a simple graph , Aigner and
Fromme's lemma states that one cop can guard a shortest path in the sense that
the robber cannot enter this path without getting caught after finitely many
steps. In this paper, we extend Aigner and Fromme's lemma to cover a larger
family of graphs and give metric characterizations of these graphs. In
particular, we show that a generalization of block graphs, namely vertebrate
graphs, are 1-guardable. We use this result to give the cop number of some
special class of multi-layer generalized Peterson graphs.Comment: fixing typo
On the size-Ramsey number of tight paths
For any and , the -color size-Ramsey number of a -uniform hypergraph is the smallest
integer such that there exists a -uniform hypergraph on
edges such that any coloring of the edges of with colors
yields a monochromatic copy of . Let
denote the -uniform tight path on vertices. Dudek, Fleur, Mubayi and
R\H{o}dl showed that the size-Ramsey number of tight paths where
. In this paper, we improve their bound
by showing that
for all and .Comment: 9 page
On the cover Tur\'an number of Berge hypergraphs
For a fixed set of positive integers , we say is an
-uniform hypergraph, or -graph, if the cardinality of each edge belongs
to . For a graph , a hypergraph is called a
Berge-, denoted by , if there exists a bijection such that for every , . In this
paper, we define a variant of Tur\'an number in hypergraphs, namely the cover
Tur\'an number, denoted as , as the maximum number of edges
in the shadow graph of a Berge- free -graph on vertices. We show a
general upper bound on the cover Tur\'an number of graphs and determine the
cover Tur\'an density of all graphs when the uniformity of the host hypergraph
equals to .Comment: 14 page
Equivalence of L0 and L1 Minimizations in Sudoku Problem
Sudoku puzzles can be formulated and solved as a sparse linear system of
equations. This problem is a very useful example for the Compressive Sensing
(CS) theoretical study. In this study, the equivalence of Sudoku puzzles L0 and
L1 minimizations is analyzed. In particular, 17-clue (smallest number of clues)
uniquely completable puzzles with sparse optimization algorithms are studied
and divided into two types, namely, type-I and -II puzzles. The solution of L1
minimization for the type-I puzzles is unique, and the sparse optimization
algorithms can solve all of them exactly. By contrast, the solution of L1
minimization is not unique for the type-II puzzles, and the results of
algorithms are incorrect for all these puzzles. Each empty cell for all type-II
puzzles is examined. Results show that some cells can change the equivalence of
L0 and L1 minimizations. These results may be helpful for the study of
equivalence of L0 and L1 norm minimization in CS
The extremal -spectral radius of Berge-hypergraphs
Let be a graph. We say that a hypergraph is a Berge- if there is a
bijection such that for all . For any -uniform hypergraph and a real number , the
-spectral radius of is defined as In this
paper, we study the -spectral radius of Berge- hypergraphs. We determine
the -uniform hypergraphs with maximum -spectral radius for
among Berge- hypergraphs when is a path, a cycle or a star.Comment: 15 page
On a hypergraph probabilistic graphical model
We propose a directed acyclic hypergraph framework for a probabilistic
graphical model that we call Bayesian hypergraphs. The space of directed
acyclic hypergraphs is much larger than the space of chain graphs. Hence
Bayesian hypergraphs can model much finer factorizations than Bayesian networks
or LWF chain graphs and provide simpler and more computationally efficient
procedures for factorizations and interventions. Bayesian hypergraphs also
allow a modeler to represent causal patterns of interaction such as Noisy-OR
graphically (without additional annotations). We introduce global, local and
pairwise Markov properties of Bayesian hypergraphs and prove under which
conditions they are equivalent. We define a projection operator, called shadow,
that maps Bayesian hypergraphs to chain graphs, and show that the Markov
properties of a Bayesian hypergraph are equivalent to those of its
corresponding chain graph. We extend the causal interpretation of LWF chain
graphs to Bayesian hypergraphs and provide corresponding formulas and a
graphical criterion for intervention
Concentration inequalities in spaces of random configurations with positive Ricci curvatures
In this paper, we prove an Azuma-Hoeffding-type inequality in several
classical models of random configurations, including the Erd\H{o}s-R\'enyi
random graph models and , the random -out(in)-regular
directed graphs, and the space of random permutations. The main idea is using
Ollivier's work on the Ricci curvature of Markov chairs on metric spaces. Here
we give a cleaner form of such concentration inequality in graphs. Namely, we
show that for any Lipschitz function on any graph (equipped with an ergodic
random walk and thus an invariant distribution ) with Ricci curvature at
least , we have Comment: 22 page
On the cover Ramsey number of Berge hypergraphs
For a fixed set of positive integers , we say is an
-uniform hypergraph, or -graph, if the cardinality of each edge belongs
to . An -graph is \emph{covering} if every vertex pair of
is contained in some hyperedge. For a graph , a
hypergraph is called a \textit{Berge}-, denoted by , if
there exists an injection such that for every , . In this note, we define a new type of Ramsey
number, namely the \emph{cover Ramsey number}, denoted as , as the smallest integer such that for every covering -uniform
hypergraph on vertices and every -edge-coloring
(blue and red) of , there is either a blue Berge- or a red
Berge- subhypergraph. We show that for every , there exists some
such that for any finite graphs and , . Moreover, we show that
for each positive integer and , there exists a constant
such that if is a graph on vertices with maximum degree at most ,
then .Comment: 9 page
On the size of planar graphs with positive Lin-Lu-Yau Ricci curvature
We show that if a planar graph with minimum degree at least has
positive Lin-Lu-Yau Ricci curvature on every edge, then ,
which then implies that is finite. This is an analogue of a result of DeVos
and Mohar [{\em Trans. Amer. Math. Soc., 2007}] on the size of planar graphs
with positive combinatorial curvature.Comment: 10 pages, 2 figure
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