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 T ⊆ V 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.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics