Modulo orientations and matchings in graphs

Abstract

The modulo orientation problem seeks a so-called mod (2t+1)-orientation of an undirected graph, in which the indegree is equal to outdegree under modulo 2t+1 at each vertex. Jaeger's circular flow conjecture states that every graph G with edge connectivity κ′(G)≥4t has a mod (2t+1)-orientation. Lovász et al. (2013) verified it for κ′(G)≥6t, and later Han et al. (2018) disproved Jaeger's conjecture with infinitely many counterexamples for t≥3. In this paper, we show there are essentially finitely many exceptions for graphs with a bounded matching number. More generally, for any positive integers t and s, there exists a finite family G(t,s) of graphs not admitting any mod (2t+1)-orientations, such that any graph G with κ′(G)≥2t+2 and matching number α′(G)≤s has a mod (2t+1)-orientation if and only if G cannot be contracted to an element of G(t,s). This immediately implies a Chvátal-Erdős type theorem and we additionally characterize all infinitely many graphs with κ′≥α′ but without a nowhere-zero 3-flow. Our results also indicate that the problem of seeking mod orientations for planar graphs with bounded matching number belongs to P, while for general planar graphs it is a known NP-complete problem.