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We develop four constructions for nowhere-zero 5-ows of 3-regular graphs which satisfy special structural conditions. Using these constructions we show a minimal counterexample to Tutte's 5-ow conjecture is of order 44 and therefore every bridgeless graph of nonorientable genus 5 has a nowhere-zero 5-ow. One of the structural properties is formulated in(More)
Snarks are bridgeless cubic graphs with chromatic index χ = 4. A snark G is called critical if χ (G − {v, w}) = 3, for any two adjacent vertices v and w. For any k ≥ 2 we construct cyclically 5-edge connected critical snarks G having an independent set I of at least k vertices such that χ (G − I) = 4. For k = 2 this solves a problem of Nedela andŠkoviera(More)