Graph Theory Chapter 9 Planar Graphs
... If G is a connected plane graph with p vertices, q edges, and r regions, then p - q + r = 2. pf: (by induction on q) (basis) If q = 0, then G K1; so p = 1, r =1, and p - q + r = 2. (inductive) Assume the result is true for any graph with q = k - 1 edges, where k 1. Let G be a graph with k edges. ...
... If G is a connected plane graph with p vertices, q edges, and r regions, then p - q + r = 2. pf: (by induction on q) (basis) If q = 0, then G K1; so p = 1, r =1, and p - q + r = 2. (inductive) Assume the result is true for any graph with q = k - 1 edges, where k 1. Let G be a graph with k edges. ...
Amenable Actions of Nonamenable Groups
... (Fm , H) is equivalent to amenability of the graph Γ, which can be defined as existence of a sequence {Fn } of finite subsets of Γ with the property that |∂Fn |/|Fn | → 0 as n → ∞, where ∂Fn is the boundary of Fn (for amenability of graphs see [CSGH99]). One of properties that insure amenability of ...
... (Fm , H) is equivalent to amenability of the graph Γ, which can be defined as existence of a sequence {Fn } of finite subsets of Γ with the property that |∂Fn |/|Fn | → 0 as n → ∞, where ∂Fn is the boundary of Fn (for amenability of graphs see [CSGH99]). One of properties that insure amenability of ...
