Modulo 5-orientations and degree sequences.

In connection to the 5-flow conjecture, a modulo 5-orientation of a graph G is an orientation of G such that the indegree is congruent to outdegree modulo 5 at each vertex. Jaeger conjectured that every 9-edge-connected multigraph admits a modulo 5-orientation, whose truth would imply Tutte’s 5-flow conjecture. In this paper, we study the problem of modulo 5-orientation for given multigraphic degree sequences. We prove that a multigraphic degree sequence d=(d1,…,dn) has a realization G with a modulo 5-orientation if and only if di≠1,3 for each i. In addition, we show that every multigraphic sequence d=(d1,…,dn) with min1≤i≤ndi≥9 has a 9-edge-connected realization which admits a modulo 5-orientation for every possible boundary function. This supports the above mentioned conjecture of Jaeger.