1,840 research outputs found
Characterization Of A Class Of Graphs Related To Pairs Of Disjoint Matchings
For a given graph consider a pair of disjoint matchings the union of which
contains as many edges as possible. Furthermore, consider the relation of the
cardinalities of a maximum matching and the largest matching in those pairs. It
is known that this relation does not exceed 5/4 for any graph. We characterize
the class of graphs for which this relation is precisely 5/4. Our
characterization implies that these graphs contain a spanning subgraph, every
component of which is the minimal graph of this class.Comment: 33 pages, 10 figure
Pairs of disjoint matchings and related classes of graphs
For a finite graph , we study the maximum -edge colorable subgraph
problem and a related ratio , where is the
matching number of , and is the size of the largest matching in any
pair of disjoint matchings maximizing (equivalently,
forming a maximum -edge colorable subgraph). Previously, it was shown that
, and the class of graphs
achieving was completely characterized. We show here that any
rational number between and can be achieved by a connected
graph. Furthermore, we prove that every graph with ratio less than must
admit special subgraphs
Spanning trees of 3-uniform hypergraphs
Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting
spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done
in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and
related to the class of Pfaffian graphs. We prove a complexity result for
recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian
3-graphs -- one of these is given by a forbidden subgraph characterization
analogous to Little's for bipartite Pfaffian graphs, and the other consists of
a class of partial Steiner triple systems for which the property of being
3-Pfaffian can be reduced to the property of an associated graph being
Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are
not 3-Pfaffian, none of which can be reduced to any other by deletion or
contraction of triples.
We also find some necessary or sufficient conditions for the existence of a
spanning tree of a 3-graph (much more succinct than can be obtained by the
currently fastest polynomial-time algorithm of Gabow and Stallmann for finding
a spanning tree) and a superexponential lower bound on the number of spanning
trees of a Steiner triple system.Comment: 34 pages, 9 figure
On disjoint matchings in cubic graphs
For and a cubic graph let denote the maximum number
of edges that can be covered by matchings. We show that and . Moreover, it turns out that
.Comment: 41 pages, 8 figures, minor chage
Bi-Criteria and Approximation Algorithms for Restricted Matchings
In this work we study approximation algorithms for the \textit{Bounded Color
Matching} problem (a.k.a. Restricted Matching problem) which is defined as
follows: given a graph in which each edge has a color and a profit
, we want to compute a maximum (cardinality or profit)
matching in which no more than edges of color are
present. This kind of problems, beside the theoretical interest on its own
right, emerges in multi-fiber optical networking systems, where we interpret
each unique wavelength that can travel through the fiber as a color class and
we would like to establish communication between pairs of systems. We study
approximation and bi-criteria algorithms for this problem which are based on
linear programming techniques and, in particular, on polyhedral
characterizations of the natural linear formulation of the problem. In our
setting, we allow violations of the bounds and we model our problem as a
bi-criteria problem: we have two objectives to optimize namely (a) to maximize
the profit (maximum matching) while (b) minimizing the violation of the color
bounds. We prove how we can "beat" the integrality gap of the natural linear
programming formulation of the problem by allowing only a slight violation of
the color bounds. In particular, our main result is \textit{constant}
approximation bounds for both criteria of the corresponding bi-criteria
optimization problem
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
Graphs Identified by Logics with Counting
We classify graphs and, more generally, finite relational structures that are
identified by C2, that is, two-variable first-order logic with counting. Using
this classification, we show that it can be decided in almost linear time
whether a structure is identified by C2. Our classification implies that for
every graph identified by this logic, all vertex-colored versions of it are
also identified. A similar statement is true for finite relational structures.
We provide constructions that solve the inversion problem for finite
structures in linear time. This problem has previously been shown to be
polynomial time solvable by Martin Otto. For graphs, we conclude that every
C2-equivalence class contains a graph whose orbits are exactly the classes of
the C2-partition of its vertex set and which has a single automorphism
witnessing this fact.
For general k, we show that such statements are not true by providing
examples of graphs of size linear in k which are identified by C3 but for which
the orbit partition is strictly finer than the Ck-partition. We also provide
identified graphs which have vertex-colored versions that are not identified by
Ck.Comment: 33 pages, 8 Figure
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