Matchings in Countable Graphs

Abstract

Tutte [9] has given necessary and sufficient conditions for a finite graph to have a perfect matching. Different proofs are given by Brualdi [1] and Gallai [2; 3]. The shortest proof of Tutte's theorem is due to Lovasz [5]. In another paper [10] Tutte extended his conditions for a perfect matching to locally finite graphs. In [4] Kaluza gave a condition on arbitrary graphs which is entirely different from Tutte's.

Strongly Maximal Matchings in Infinite Graphs

“…An even stronger notion than strong maximality of a matching in a graph is that of having (inclusion-wise) maximal support. Similarly to the proof of Lemma 3.1 it is possible to show: In [7] the following stronger version of Theorem 1.3 was proved for countable graphs:…”

“…In particular, each U can be dominated by at most one vertex. Moreover, (7) implies (i) and (ii): Indeed, consider any set U ∈ U i . For every v ∈ U , the symmetric difference of M i with the M i -alternating x U -v path of even length in G ′ i [U ] is a matching of U − v, which shows (i).…”

“…If a set X ⊂ V (G ′ i ) contains the vertices of such a path, we say that x dominates y via X. We claim that (7) For every U ∈ U i there is a vertex x U ∈ U that dominates every v ∈ U via U . For a vertex x U as in (7) we say that x U dominates U .…”

“…We claim that (7) For every U ∈ U i there is a vertex x U ∈ U that dominates every v ∈ U via U . For a vertex x U as in (7) we say that x U dominates U . Clearly (7) implies that every vertex v in U − x U is matched by M i to another vertex in U − x U (namely, to its predecessor in the M i -alternating x U -v path in G ′ i [U ] of even length), while x U either lies in X ′ i (i.e.…”

“…For a vertex x U as in (7) we say that x U dominates U . Clearly (7) implies that every vertex v in U − x U is matched by M i to another vertex in U − x U (namely, to its predecessor in the M i -alternating x U -v path in G ′ i [U ] of even length), while x U either lies in X ′ i (i.e. is unmatched by M i ) or is matched by M i to a vertex outside U .…”