Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V,E) having a k-edge-connected T-odd orientation for every subset with |E| + |T| even. (T-odd orientation: the in-degree of v is odd precisely if v is in T.) As a corollary, we obtain that every (2k)-edge-connected graph with |V| + |E| even has a (k-1)-edge-connected orientation in which the in-degree of every node is odd. Along the way, a structural characterization will be given for digraphs with a root-node s having k edge-disjoint paths from s to every node and k-1 edge-disjoint paths from every node to s.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received December 14, 1998/Revised January 12, 2001
RID="*"
ID="*" Supported by the Hungarian National Foundation for Scientific Research, OTKA T029772. Part of research was done while this author was visiting EPFL, Lausanne, June, 1998.
RID="†"
ID="†" Supported by the Hungarian National Foundation for Scientific Research, OTKA T029772 and OTKA T030059.
Rights and permissions
About this article
Cite this article
Frank, A., Király, Z. Graph Orientations with Edge-connection and Parity Constraints. Combinatorica 22, 47–70 (2002). https://doi.org/10.1007/s004930200003
Issue Date:
DOI: https://doi.org/10.1007/s004930200003