TUTTES 5-FLOW CONJECTURE FOR GRAPHS OF NONORIENTABLE GENUS-5

We develop four constructions for nowhere-zero 5-flows of 3-regular gr aphs that satisfy special structural conditions. Using these construct ions we show a minimal counterexample to Tutte's 5-Flow Conjecture is of order greater than or equal to 44 and therefore every bridgeless gr aph of nonorientable genus less than or equal to 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 grap h G by contracting the cycles of a 2-factor F in G. (C) 1996 John Wile y & Sons, Inc.