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Chain Packings and Odd Subtree Packings
Chain Packings and Odd Subtree Packings
Garth Isaak
Department of Mathematics and Computer Science
Dartmouth College, Hanover, NH
1992
Abstract
A chain packing H in a graph is a subgraph satisfying given degree con-
straints at the vertices. Its size is the number of odd degree vertices in the
subgraph. An odd subtree packing is a chain packing which is a forest in
which all non-isolated vertices have odd degree in the forest. We show that
for a given graph and degree constraints, the size of a maximum chain pack-
ing and a maximum odd subtree packing are the same but the same does not
hold for a version in which the sum of given weights on the odd degree ver-
tices is to be maximized. We also note a reduction to weighted capacitated
b-matching for finding a maximum size chain packing, maximum size odd
subtree packing and maximum weight chain packing. The main result of
this note is the proof that a min-max formula generalizing the Berge-Tutte
formula for matching holds for chain packing.
1 Introduction
Edge disjoint packings of chains as an extension of matching have been
studied by deWerra [14, 15, 16], deWerra and Pulleyblank [13] and deWerra
and Roberts [17]. In [14, 15, 16, 13] the chains have odd length. Chain
Packings of general length are studied in [17]. A chain packing H in a
graph G is a subgraph of G satisfying given degree constraints. Its size
is the number of vertices with odd degree in H. We will examine chain
packings and a closely related notion of odd subtree packings. An odd subtree
packing F is a chain packing for which each non-trivial component is a tree
containing no even degree vertices. Its size is the number of odd degree
vertices in F . Thus, odd subtree packing can be viewed as a generalization
of matching in which matched edges are replaced by odd subtrees satisfying
certain degree constraints. Related, but distinct problems of packing with
the subgraphs drawn from a fixed family and finding subgraphs satisfying
degree constraints have
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