Spanning Graphs and the Axiom of Choice

A b s t r a c t. We show in set-theory ZF that the axiom of choice is equivalent to the statement every bipartite connected graph has a spanning sub-graph omitting some complete finite bipartite graph K n,n , and in particular it is equivalent to the fact that every connected graph has a spanning cycle-free graph (possibly non connected). We consider simple undirected loop-free graphs. A forest is a graph with no cycles, a tree is a connected forest. A graph G is a sub-graph of G if all its edges (and vertices) belong to G ; say that such a sub-graph is spanning if every vertex of G belongs to an edge of G .