224 research outputs found
Ear-Slicing for Matchings in Hypergraphs
We study when a given edge of a factor-critical graph is contained in a
matching avoiding exactly one, pregiven vertex of the graph. We then apply the
results to always partition the vertex-set of a -regular, -uniform
hypergraph into at most one triangle (hyperedge of size ) and edges (subsets
of size of hyperedges), corresponding to the intuition, and providing new
insight to triangle and edge packings of Cornu\'ejols' and Pulleyblank's. The
existence of such a packing can be considered to be a hypergraph variant of
Petersen's theorem on perfect matchings, and leads to a simple proof for a
sharpening of Lu's theorem on antifactors of graphs
How many matchings cover the nodes of a graph?
Given an undirected graph, are there matchings whose union covers all of
its nodes, that is, a matching--cover? A first, easy polynomial solution
from matroid union is possible, as already observed by Wang, Song and Yuan
(Mathematical Programming, 2014). However, it was not satisfactory neither from
the algorithmic viewpoint nor for proving graphic theorems, since the
corresponding matroid ignores the edges of the graph.
We prove here, simply and algorithmically: all nodes of a graph can be
covered with matchings if and only if for every stable set we have
. When , an exception occurs: this condition is not
enough to guarantee the existence of a matching--cover, that is, the
existence of a perfect matching, in this case Tutte's famous matching theorem
(J. London Math. Soc., 1947) provides the right `good' characterization. The
condition above then guarantees only that a perfect -matching exists, as
known from another theorem of Tutte (Proc. Amer. Math. Soc., 1953).
Some results are then deduced as consequences with surprisingly simple
proofs, using only the level of difficulty of bipartite matchings. We give some
generalizations, as well as a solution for minimization if the edge-weights are
non-negative, while the edge-cardinality maximization of matching--covers
turns out to be already NP-hard.
We have arrived at this problem as the line graph special case of a model
arising for manufacturing integrated circuits with the technology called
`Directed Self Assembly'.Comment: 10 page
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