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Partitions into odd chains.(English)Zbl 0616.90015

An odd chain packing (OCP) in a graph is an edge-disjoint collection P of odd length chains (length of a chain is the number of edges in it) and possibly nonelementary even cycles, such that no two chains in P have a common endpoint. The OCP partition number of a graph \(G=(V,E)\) is the smallest K for which there exists a partion of E into K OCPs.
Necessary and sufficient conditions for a bipartite multigraph to be partitioned into K OCPs are given. These conditions are then converted into a min-max formula for the OCP partition number. The author also proves that for any graph, as long as \(K\geq\) its OCP partition number, its edge set can be partitioned into K OCPs for which the number of chains in the various OCPs differ by at most one.
Reviewer: K.G.Murty

MSC:

90B10 Deterministic network models in operations research
90C27 Combinatorial optimization
05C38 Paths and cycles
90C35 Programming involving graphs or networks

Cite

References:

[1]C. Berge,Graphes (Dunod, Paris, 1983).
[2]R.B. Eggleton and D.K. Skilton, ”Chains decompositions of graphs”, in:Graph Theory, Singapore 1983, Lecture Notes in Mathematics 1073, (Springer, Berlin, 1984), Part I, pp. 294–306, part II, pp. 307–323.
[3]J. Folkman and D.R. Fulkerson, ”Edge colorings in bipartite graphs,” in: R. C. Bose and T.A. Dowling, eds.,Combinatorial Mathematics and their Applications (University of North Carolina Press, Chapel Hill, 1969) pp. 561–577. ·Zbl 0204.57002
[4]L. Lovasz, ”On covering of graphs”, in: P. Erdös and G. Katona, eds.,Theory of Graphs (Academic Press, New York, 1968) pp. 231–236.
[5]B. Rotschild, ”The decomposition of graphs into a finite number of paths”Canadian Jounal of Mathematics 17 (1965) 468–479. ·Zbl 0129.40003 ·doi:10.4153/CJM-1965-046-1
[6]P. Seymour, ”On odd cuts and plane multicommodity flows,”Proceedings of the London Mathematical Society 3 42 (1981) 178–192. ·Zbl 0447.90026 ·doi:10.1112/plms/s3-42.1.178
[7]D. de Werra, ”Variation on a theorem of König”,Discrete Mathematics 51 (1984) 319–321. ·Zbl 0545.05049 ·doi:10.1016/0012-365X(84)90015-3
[8]D. de Werra, ”Node coverings with odd chains”,Journal of Graph Theory 10 (1986) 177–185. ·Zbl 0651.05055 ·doi:10.1002/jgt.3190100206
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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