Harmonic functions on planar and almost planar graphs and ...

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584 I. Benjamini, O. Schrammg : |G| →D, by letting g map each I e into the path f ∗ (e). It is easy to verifythat the number of crossings for each edge in the drawing is bounded by C times2κ + 1 times the maximal degree of G ◦ . Hence there is a drawing of G in theplane in which the number of crossings in any edge is bounded.Now suppose that there is a drawing g : |G| →R 2 of G in the plane so thatevery edge e ∈ E has at most κ
<strong>Harmonic</strong> <strong>functions</strong> 5857.1. Theorem Let G =(V,E)be a finite genus, connected, locally finite graph.Then G can be drawn in the plane with finitely many crossings. In particular, ifG has bounded valence, then it is almost planar.Proof. Let S be a closed (compact, without boundary) surface such that G embeddsin S . We assume, with no loss of generality, that G is a triangulation of adomain D ⊂ S . Recall that the genus of any boundaryless surface X is the maximumcardinality of a collection {γ 1 ,...,γ g } of disjoint simple closed curvesin X such that X −∪ j γ j is connected. Since the genus of D is bounded by thegenus of S , it is finite. Hence there is a finite collection of edges E 0 ⊂ E suchthat any simple closed path in G − E 0 separates D. Since G is a triangulationof D, this implies that there is no simple closed non-separating path in D − E 1 ,where E 1 is the set of all edges in G that share some vertex with some edgein E 0 . Therefore every component of D − E 1 has zero genus, and is planar. Inparticular, G − E 1 is planar. Let E ′ ⊂ E 1 be a minimal set of edges subject tothe property that G − E ′ is planar. (Here, G − E ′ contains all the vertices of G.)Clearly, G − E ′ is connected, and E ′ is finite.Let f be an embedding of G −E ′ in the plane so that for every edge or vertexj of G − E ′ there is an open set A containing f (j ) that intersects at most finitelymany of the vertices and edges of f (G − E ′ ). Now suppose that e is some edgein E ′ , and let v, w be its vertices. Since G − E ′ is connected, there is a path γin f (G − E ′ ) joining f (v) and f (w). We may define f (e) to be a simple path thatfollows alongside γ and intersects f (G − E ′ ) in finitely many points. Similarly,all the edges in E ′ may be drawn in such a way that there are finitely manycrossings. ⊓⊔Almost planar graphs and separation propertiesIt is known [18] that any planar graph G =(V,E) has the √ n separation property.That is, there are c 0 < ∞, c 1 < 1, so that for any finite set W ⊂ V of nvertices, there is a subset W ′√⊂ W containing at most c 0 n vertices, suchthat any connected subset of W − W ′ has at most c 1 n vertices. Clearly almostplanar graphs have this √ n separation property too. A natural question is whether√ n separation is equivalent to almost planarity. And if not, do transient graphswith the √ n separtion property always admit non constant bounded harmonic<strong>functions</strong>?Examples. Let T be a tree, and denote by T × Z the product of T by Z. InLemma 7.2 below, we show that T × Z has the √ n separation property. Couplingshows that any bounded harmonic function h on T ×Z must be constant on everyfibre of the form {t}×Z. Hence the only harmonic Dirichlet <strong>functions</strong> on T × Zare the constants. By Theorem 1.9 it then follows that T × Z is not almost planar,if it is transient. In particular, when T is the binary tree, T × Z is a graph havingthe √ n separation property, which is not almost planar. Hence, √ n separationdoes not imply almost planarity.
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