Chain Packings and Odd Subtree Packings.pdfVIP

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