Nowhere-zero 3-flows and modulo k-orientations

  • Authors:

  • László Miklós Lovász;Carsten Thomassen;Yezhou Wu;Cun-Quan Zhang

  • Affiliations:

  • Trinity College, University of Cambridge, Cambridge, CB2 1TQ, United Kingdom;Department of Mathematics, Technical University of Denmark, DK-2800 Lyngby, Denmark and King Abdulaziz University, Jeddah, Saudi Arabia;School of Mathematics and Statistics, Jiangsu Normal University, Jiangsu, 221116, China and Department of Mathematics, West Virginia University, Morgantown, WV 26506, United States;Department of Mathematics, West Virginia University, Morgantown, WV 26506, United States

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2013

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Abstract

The main theorem of this paper provides partial results on some major open problems in graph theory, such as Tutte@?s 3-flow conjecture (from the 1970s) that every 4-edge connected graph admits a nowhere-zero 3-flow, the conjecture of Jaeger, Linial, Payan and Tarsi (1992) that every 5-edge-connected graph is Z"3-connected, Jaeger@?s circular flow conjecture (1984) that for every odd natural number k=3, every (2k-2)-edge-connected graph has a modulo k-orientation, etc. It was proved recently by Thomassen that, for every odd number k=3, every (2k^2+k)-edge-connected graph G has a modulo k-orientation; and every 8-edge-connected graph G is Z"3-connected and admits therefore a nowhere-zero 3-flow. In the present paper, Thomassen@?s method is refined to prove the following: For every odd numberk=3, every(3k-3)-edge-connected graph has a modulo k-orientation. As a special case of the main result, every 6-edge-connected graph isZ"3-connected and admits therefore a nowhere-zero 3-flow. Note that it was proved by Kochol (2001) that it suffices to prove the 3-flow conjecture for 5-edge-connected graphs.