Abstract
An odd chain packingP in a graph is an edge-disjoint collection of odd length chains (and even cycles) such that each node is the endpoint of at most one chain ofP. We define the ocp partition number of a graphG as the smallestk such that the edge setG can be partitioned intok odd chain packings. A min-max theorem is given for bipartite graphs: it is an analogue of the second theorem of König. Some remarks on the corresponding covering problem are given and some other odd chain problems are mentioned.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Berge,Graphes (Dunod, Paris, 1983).
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.
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.
L. Lovasz, “On covering of graphs”, in: P. Erdös and G. Katona, eds.,Theory of Graphs (Academic Press, New York, 1968) pp. 231–236.
B. Rotschild, “The decomposition of graphs into a finite number of paths”Canadian Jounal of Mathematics 17 (1965) 468–479.
P. Seymour, “On odd cuts and plane multicommodity flows,”Proceedings of the London Mathematical Society 3 42 (1981) 178–192.
D. de Werra, “Variation on a theorem of König”,Discrete Mathematics 51 (1984) 319–321.
D. de Werra, “Node coverings with odd chains”,Journal of Graph Theory 10 (1986) 177–185.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
de Werra, D. Partitions into odd chains. Mathematical Programming 37, 41–50 (1987). https://doi.org/10.1007/BF02591682
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02591682