Graph Orientations with Edge-connection and Parity Constraints

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.

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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.

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Frank, A., Király, Z. Graph Orientations with Edge-connection and Parity Constraints. Combinatorica 22, 47–70 (2002). https://doi.org/10.1007/s004930200003

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  • AMS Subject Classification (2000) Classes:  05C75, 05C40