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We develop four constructions for nowhere-zero 5-flows of 3-regular graphs that satisfy special structural conditions. Using these constructions we show a minimal counterexample to Tutte's 5-Flow Conjecture is of order 244 and therefore every bridgeless graph of nonorientable genus 5 5 has a nowhere-zero 5-flow. One of the structural properties is formulated in terms of the structure of the multigraph G(F) obtained from a given 3-regular graph G by contracting the cycles of a 2-factor F in G.

Tutte's 5-flow conjecture for graphs of nonorientable genus 5