Abstract
A cycle of a graph G is a set C ⊑ E(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.
Supported by Project LN00A056 and 1M0021620808 of the Czech Ministery of Education.
Supported by Barrande 02887WD P.A.I. Franco-Tchèque.
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DeVos, M., Nešetřil, J., Raspaud, A. (2006). On Edge-maps whose Inverse Preserves Flows or Tensions. In: Bondy, A., Fonlupt, J., Fouquet, JL., Fournier, JC., Ramírez Alfonsín, J.L. (eds) Graph Theory in Paris. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7400-6_10
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