Abstract
A graph G = (V, E) admits a nowhere-zero k-flow if there exists an orientation H = (V, A) of G and an integer flow \({\varphi:A \to \mathbb{Z}}\) such that for all \({a \in A, 0 < |\varphi(a)| < k}\). Tutte conjectured that every bridgeless graphs admits a nowhere-zero 5-flow. A (1,2)-factor of G is a set \({F \subseteq E}\) such that the degree of any vertex v in the subgraph induced by F is 1 or 2. Let us call an edge of G, F-balanced if either it belongs to F or both its ends have the same degree in F. Call a cycle of G F-even if it has an even number of F-balanced edges. A (1,2)-factor F of G is even if each cycle of G is F-even. The main result of the paper is that a cubic graph G admits a nowhere-zero 5-flow if and only if G has an even (1,2)-factor.
Similar content being viewed by others
References
Celmins, U.A.: On cubic graphs that do not have an edge-3-coloring, PhD thesis, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada (2004)
Jaeger, F.: Selected topics in graph theory, vol. 3, pp. 71–95. Academic Press, San Diego
Kochol M.: Reduction of the 5-flow conjecture to cyclically 6-edge-connected snarks. J. Combin. Theory Ser. B 90, 139–145 (2004)
Kochol M.: Restrictions on smallest counterexamples to the 5-flow conjecture. Combinatorica 26, 83–89 (2006)
Seymour P.D.: Nowhere-zero 6-flows. J. Combin. Theory Ser. B 30, 130–135 (1981)
Steffen E.: Tutte’s 5-flow conjecture for graphs of nonorientable genus 5. J. Graph Theory 22, 309–319 (1996)
Tutte W.T.: On the imbedding of linear graphs in surfaces. Proc. Lond. Math. Soc. 51(2), 474–483 (1949)
Tutte W.T.: A contribution to the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954)
Zhang, C.Q: Integer flows and cycle covers of graphs. In: Pure and Applied Mathematics, New York
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by CONICYT doctoral grant, Proyecto Anillo de Redes ACT-08, FONDAP and BASAL-CMM projects.
Rights and permissions
About this article
Cite this article
Matamala, M., Zamora, J. Nowhere-Zero 5-Flows and Even (1,2)-Factors. Graphs and Combinatorics 29, 609–616 (2013). https://doi.org/10.1007/s00373-011-1119-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-011-1119-x