Nowhere-zero 3-flows in Cayley graphs and Sylow 2-subgroups

Authors:
Mária Nánásiová;Martin Škoviera
Affiliations:
Department of Computer Science, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia 842 48;Department of Computer Science, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia 842 48
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2009

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Abstract

Tutte's 3-Flow Conjecture suggests that every bridgeless graph with no 3-edge-cut can have its edges directed and labelled by the numbers 1 or 2 in such a way that at each vertex the sum of incoming values equals the sum of outgoing values. In this paper we show that Tutte's 3-Flow Conjecture is true for Cayley graphs of groups whose Sylow 2-subgroup is a direct factor of the group; in particular, it is true for Cayley graphs of nilpotent groups. This improves a recent result of Poto驴nik et al. (Discrete Math. 297:119---127, 2005) concerning nowhere-zero 3-flows in abelian Cayley graphs.