239 research outputs found
On the Core of a Unicyclic Graph
A set S is independent in a graph G if no two vertices from S are adjacent.
By core(G) we mean the intersection of all maximum independent sets. The
independence number alpha(G) is the cardinality of a maximum independent set,
while mu(G) is the size of a maximum matching in G. A connected graph having
only one cycle, say C, is a unicyclic graph. In this paper we prove that if G
is a unicyclic graph of order n and n-1 = alpha(G) + mu(G), then core(G)
coincides with the union of cores of all trees in G-C.Comment: 8 pages, 5 figure
Computing Unique Maximum Matchings in O(m) time for Konig-Egervary Graphs and Unicyclic Graphs
Let alpha(G) denote the maximum size of an independent set of vertices and
mu(G) be the cardinality of a maximum matching in a graph G. A matching
saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number
of vertices of G, then it is called a Konig-Egervary graph. A graph is
unicyclic if it has a unique cycle.
In 2010, Bartha conjectured that a unique perfect matching, if it exists, can
be found in O(m) time, where m is the number of edges.
In this paper we validate this conjecture for Konig-Egervary graphs and
unicylic graphs. We propose a variation of Karp-Sipser leaf-removal algorithm
(Karp and Spiser, 1981), which ends with an empty graph if and only if the
original graph is a Konig-Egervary graph with a unique perfect matching
obtained as an output as well.
We also show that a unicyclic non-bipartite graph G may have at most one
perfect matching, and this is the case where G is a Konig-Egervary graph.Comment: 10 pages, 5 figure
Problems on Matchings and Independent Sets of a Graph
Let be a finite simple graph. For , the difference of
, where is the neighborhood of and is called the critical difference of . is
called a critical set if equals the critical difference and ker is
the intersection of all critical sets. It is known that ker is an
independent (vertex) set of . diadem is the union of all critical
independent sets. An independent set is an inclusion minimal set with if no proper subset of has positive difference.
A graph is called K\"onig-Egerv\'ary if the sum of its independence
number () and matching number () equals . It is
known that bipartite graphs are K\"onig-Egerv\'ary.
In this paper, we study independent sets with positive difference for which
every proper subset has a smaller difference and prove a result conjectured by
Levit and Mandrescu in 2013. The conjecture states that for any graph, the
number of inclusion minimal sets with is at least the critical
difference of the graph. We also give a short proof of the inequality
kerdiadem (proved by Short in 2016).
A characterization of unicyclic non-K\"onig-Egerv\'ary graphs is also
presented and a conjecture which states that for such a graph , the critical
difference equals , is proved.
We also make an observation about ker using Edmonds-Gallai Structure
Theorem as a concluding remark.Comment: 18 pages, 2 figure
On the probability of planarity of a random graph near the critical point
Consider the uniform random graph with vertices and edges.
Erd\H{o}s and R\'enyi (1960) conjectured that the limit
\lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} exists
and is a constant strictly between 0 and 1. \L uczak, Pittel and Wierman (1994)
proved this conjecture and Janson, \L uczak, Knuth and Pittel (1993) gave lower
and upper bounds for this probability.
In this paper we determine the exact probability of a random graph being
planar near the critical point . For each , we find an exact
analytic expression for
In particular, we obtain .
We extend these results to classes of graphs closed under taking minors. As
an example, we show that the probability of being
series-parallel converges to 0.98003.
For the sake of completeness and exposition we reprove in a concise way
several basic properties we need of a random graph near the critical point.Comment: 10 pages, 1 figur
On rigidity, orientability and cores of random graphs with sliders
Suppose that you add rigid bars between points in the plane, and suppose that
a constant fraction of the points moves freely in the whole plane; the
remaining fraction is constrained to move on fixed lines called sliders. When
does a giant rigid cluster emerge? Under a genericity condition, the answer
only depends on the graph formed by the points (vertices) and the bars (edges).
We find for the random graph the threshold value of
for the appearance of a linear-sized rigid component as a function of ,
generalizing results of Kasiviswanathan et al. We show that this appearance of
a giant component undergoes a continuous transition for and a
discontinuous transition for . In our proofs, we introduce a
generalized notion of orientability interpolating between 1- and
2-orientability, of cores interpolating between 2-core and 3-core, and of
extended cores interpolating between 2+1-core and 3+2-core; we find the precise
expressions for the respective thresholds and the sizes of the different cores
above the threshold. In particular, this proves a conjecture of Kasiviswanathan
et al. about the size of the 3+2-core. We also derive some structural
properties of rigidity with sliders (matroid and decomposition into components)
which can be of independent interest.Comment: 32 pages, 1 figur
Sampling Random Colorings of Sparse Random Graphs
We study the mixing properties of the single-site Markov chain known as the
Glauber dynamics for sampling -colorings of a sparse random graph
for constant . The best known rapid mixing results for general graphs are in
terms of the maximum degree of the input graph and hold when
for all . Improved results hold when for
graphs with girth and sufficiently large where is the root of ; further improvements on
the constant hold with stronger girth and maximum degree assumptions.
For sparse random graphs the maximum degree is a function of and the goal
is to obtain results in terms of the expected degree . The following rapid
mixing results for hold with high probability over the choice of the
random graph for sufficiently large constant~. Mossel and Sly (2009) proved
rapid mixing for constant , and Efthymiou (2014) improved this to linear
in~. The condition was improved to by Yin and Zhang (2016) using
non-MCMC methods. Here we prove rapid mixing when where
is the same constant as above. Moreover we obtain
mixing time of the Glauber dynamics, while in previous rapid mixing
results the exponent was an increasing function in . As in previous results
for random graphs our proof analyzes an appropriately defined block dynamics to
"hide" high-degree vertices. One new aspect in our improved approach is
utilizing so-called local uniformity properties for the analysis of block
dynamics. To analyze the "burn-in" phase we prove a concentration inequality
for the number of disagreements propagating in large blocks
- …