# Antichains in the cycle continuous order

If , are graphs, a function is called *cycle-continuous* if the pre-image of every element of the (binary) cycle space of is a member of the cycle space of .

**Problem**Does there exist an infinite set of graphs so that there is no cycle continuous mapping between and whenever ?

The definition of a cycle-continuous mapping is based on some work of Jaeger, and the most interesting question on this subject is undoubtedly Jaeger's Petersen coloring conjecture.

Let us define a relation on the set of all finite graphs with at least one edge by the rule if there is a cycle-continuous mapping from to . It is not difficult to verify that is a quasi order (reflexive and transitive). In this order, every Eulerian graph dominates every other graph, and every graph with a cut edge is dominated by every other graph.

Let be the graph on two vertices with parallel edges. Then with all the inequalities strict, so this sequence is an infinite chain. Very little else seems to be known about this order. In particular, the problem highlighted above - does there exist an infinite antichain? remains open.

## Bibliography

* indicates original appearance(s) of problem.