MacLane's planarity criterion for locally finite graphs

Abstract

MacLane's planarity criterion states that a finite graph is planar if and only if its cycle space has a basis B such that every edge is contained in at most two members of B. Solving a problem of Wagner [Graphentheorie, Bibliographisches Institut, Mannheim, 1970], we show that the topological cycle space introduced recently by Diestel and Kühn allows a verbatim generalisation of MacLane's criterion to locally finite graphs. This then enables us to extend Kelmans' planarity criterion as well.

Eulerian edge sets in locally finite graphs

“…By (3), it holds that the smallest k * for which there is an infinite (k * ,Z)-region R of G m−2 with R ⊆ C m−2 n is at least m − 1. Now, (6) in conjunction with (5) shows that k * ≥ m. Hence, we may apply Lemma 11 to G m−2 ,Z,C m−2 n ,m (in the roles of G, Z, C, m respectively).…”

“…The topological cycle space, which is defined to be the set of all (possibly infinite but well defined) mod 2 sums of circuits, permits verbatim extensions of many theorems to locally finite graphs. Among these are MacLane's planarity criterion [3], Tutte's generating theorem [1] and Gallai's partition theorem [2]. Moreover, since the circuits of Diestel and Kühn may be infinite, it becomes possible to consider Hamilton circuits in locally finite graphs, see for instance Georgakopoulos [10] or Cui, Wang and Yu [5].…”

Hamilton circles in infinite planar graphs

“…If the boundary of a face is a finite cycle, say C , then we say that C is a facial cycle. Note that our graphs G will have nice embeddings (see Section 3) and we shall mainly work with faces of finite subgraphs of G. However, we mention that for any 2-connected locally finite plane graph G, it follows from [13] (also see [1]) that the face boundaries of G are circles in |G|.…”

Embedding a graph-like continuum in a surface

“…The work of Diestel and his students (see, for example, [2,3,6]) exploits the Freudenthal compactification of a locally finite infinite graph to study the cycle space. This is an example of a graph-like continuum.…”