Vector Flows and Integer Flows

A vector Sd-flow is a flow whose flow values are vectors in Sd, where Sd is the set of all unit vectors in Rd+1. Jain [Open Problem Garden, http://www.openproblemgarden.org/op/unit vector flows (2007)] and Thomassen [J. Combin. Theory Ser. B., 108 (2014), pp. 81–91] proved that a graph has a vector S1-flow if it has a nowhere-zero integer 3-flow. Thomassen [J. Combin. Theory Ser. B., 108 (2014), pp. 81–91] pointed out that a graph admitting a vector S1-flow may not necessarily admit a nowhere-zero integer 3-flow and presented a family of examples showing that the converse is not true. The rank of a vector S1-flow (D, f) is defined as the rank of linear space generated by all balanced vectors (v) = ( 1(v), 2(v), . . . , b(v)) for all v ∈ V (G), where i(v) is the difference between the number of outgoing edges with flow value αi from v and the number of ingoing edges with the same flow value to v. In this paper, we prove that G admits a nowhere-zero integer 3-flow if G admits a vector S1-flow with rank at most two. This result is sharp since there are examples that admit vector S1-flows with rank at least 3, but no nowhere-zero integer 3-flows.