A Cantor-Bernstein Theorem for Paths in Graphs

As every vertex of this graph has one “outgoing” and at most one “incoming” edge, each of those components is a cycle or an infinite path. In each of these paths and cycles we now select every other edge to mark the desired bijection. The Cantor-Bernstein problem, rephrased as above for graphs, has a natural generalization to paths. Let G be any graph, and let A and B be disjoint sets of vertices in G. Assume that we can find in G a set of disjoint paths from A to B that covers all of A (but not necessarily all of B), and a similar set of disjoint paths from all of B to A. Is there a set of disjoint A–B paths in G that covers both A and B? Indeed there is. This was first shown in 1969 by Pym [ 3 ], and his proof is not short. Later [ 4 ], Pym also gave a short but indirect proof, which applies the Rado Selection Principle (an equivalent of the axiom of choice) to a suitably strengthened technical statement. Further interesting background, including a deduction of Pym’s theorem from Tarski’s fixed point theorem for lattices [ 5 ], can be found in Fleiner [ 2 ]. Our aim in this note is to give two short and direct proofs. Both are elementary, and they can be read independently. Our first proof is simpler, as long as readers are at ease with sequences indexed by ordinals and how to define such sequences inductively. The second proof avoids using the Axiom of Choice, which makes it a little more technical but perhaps also more illuminating. A path in a graph G is a finite subgraph with distinct vertices v1, . . . , vk and edges v1v2, v2v3, . . . , vk−1vk. We often refer to a path by the sequence of its vertices (in this or the reverse order); it then has a natural first and a natural last vertex. If these lie in sets A and B, and no other vertex of the path lies in A∪B, we call it an A–B path. A set P of paths in G covers a set U of vertices if every vertex in U is the first or the last vertex of some path in P. Any other notation we use can be found (online) in [ 1 ].