6,782 research outputs found
On Edge-Disjoint Pairs Of Matchings
For a graph G, consider the pairs of edge-disjoint matchings whose union
consists of as many edges as possible. Let H be the largest matching among such
pairs. Let M be a maximum matching of G. We show that 5/4 is a tight upper
bound for |M|/|H|.Comment: 8 pages, 2 figures, Submitted to Discrete Mathematic
Disjoint compatibility graph of non-crossing matchings of points in convex position
Let be a set of labeled points in convex position in the plane.
We consider geometric non-intersecting straight-line perfect matchings of
. Two such matchings, and , are disjoint compatible if they do
not have common edges, and no edge of crosses an edge of . Denote by
the graph whose vertices correspond to such matchings, and two
vertices are adjacent if and only if the corresponding matchings are disjoint
compatible. We show that for each , the connected components of
form exactly three isomorphism classes -- namely, there is a
certain number of isomorphic small components, a certain number of isomorphic
medium components, and one big component. The number and the structure of small
and medium components is determined precisely.Comment: 46 pages, 30 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
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
Packing Plane Perfect Matchings into a Point Set
Given a set of points in the plane, where is even, we consider
the following question: How many plane perfect matchings can be packed into
? We prove that at least plane perfect matchings
can be packed into any point set . For some special configurations of point
sets, we give the exact answer. We also consider some extensions of this
problem
Rank Maximal Matchings -- Structure and Algorithms
Let G = (A U P, E) be a bipartite graph where A denotes a set of agents, P
denotes a set of posts and ranks on the edges denote preferences of the agents
over posts. A matching M in G is rank-maximal if it matches the maximum number
of applicants to their top-rank post, subject to this, the maximum number of
applicants to their second rank post and so on.
In this paper, we develop a switching graph characterization of rank-maximal
matchings, which is a useful tool that encodes all rank-maximal matchings in an
instance. The characterization leads to simple and efficient algorithms for
several interesting problems. In particular, we give an efficient algorithm to
compute the set of rank-maximal pairs in an instance. We show that the problem
of counting the number of rank-maximal matchings is #P-Complete and also give
an FPRAS for the problem. Finally, we consider the problem of deciding whether
a rank-maximal matching is popular among all the rank-maximal matchings in a
given instance, and give an efficient algorithm for the problem
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