Graph Theory in Paris

Part of the series Trends in Mathematics pp 109-138

On Edge-maps whose Inverse Preserves Flows or Tensions

  • Matt DeVosAffiliated withApplied Math Department, Princeton University
  • , Jaroslav NešetřilAffiliated withDepartment of Applied Mathematics (KAM) Institut for Theoretical Computer Sciences (ITI), Charles University
  • , André RaspaudAffiliated withLaBRI, Université Bordeaux I

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A cycle of a graph G is a set CE(G) so that every vertex of the graph (V (G), C) has even degree. If G,H are graphs, we define a map φ: E(G) → E(H) to be cycle-continuous if the pre-image of every cycle of H is a cycle of G. A fascinating conjecture of Jaeger asserts that every bridgeless graph has a cycle-continuous mapping to the Petersen graph. Jaeger showed that if this conjecture is true, then so is the 5-cycle-double-cover conjecture and the Fulkerson conjecture.

Cycle continuous maps give rise to a natural quasi-order ≻ on the class of finite graphs. Namely, G ≻ H if there exists a cycle-continuous mapping from G to H. The goal of this paper is to establish some basic structural properties of this (and other related) quasi-orders. For instance, we show that ≻ has antichains of arbitrarily large finite size. It appears to be an interesting question to determine if ≻ has an infinite antichain.