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Unformatted text preview: MATH 682 Notes Combinatorics and Graph Theory II 1 Planar graphs 1.1 More fun with faces We developed this idea of a face of a planar projection of a graph G , which motivated the very useful result known as Eulers Formula: if a planar projection of a connected graph has v vertices, f faces, and e edges, then v + f- e = 2. There is one useful immediate consequence of this theorem: Corollary 1. The number of faces of a planar projection connected graph G does not depend upon the choice of projection. We can develop several other properties of faces besides simply their number. Two interesting qualities of faces are their adjacencies and boundary lengths. Definition 1. If the edges bounding a face F form a closed walk of length i , the face itself is said to have length i . Note that edges separating the face from other faces will be counted once, while edges that jut into the face will be counted twice. This brings us to our second concept; namely, that two faces might share an edge. Definition 2. If the edge e lies on the boundary of two separate faces f 1 and f 2 , then f 1 and f- 2 are said to be adjacent. Definition 3. The dual graph of a planar projection of G is a graph H whose vertices are the faces of V , in which f 1 and f 2 are adjacent in H if they are adjacent in V . The dual and face lengths are useful visualization tools, but have one grievous flaw: they arent actually uniquely defined by the graph G , but only by its planar projection! Below are two planar projections of the same graph, with quite different duals ansd face-lengths. For that reason, looking at the faces of G can be inherently dangerous if we treat them as if they were uniquely defined. However, to a certain extent we can make statements about them which are true regardless of projection. For instance: Proposition 1. If a planar graph G has faces f 1 ,...,f k with respective lengths 1 ,..., k , then k i =1 i = 2 k G k . Proof. Every edge is either a boundary between two faces, in which case it lies on the boundary walk of each face, contributing 1 towards the length of each and 2 towards the total, or lies wholly surrounded by a single face, in which case it appears twice on the boundary walk for a single face, contributing twice towards its length, and thus twice towards the total. Thus, each edge is accounted for twice in a census of all face-boundaries, so the above sum is twice the number of edges in G . Proposition 2. In a planar connected graph G with 3 or more vertices, every face has length at least 3. Page 1 of 8 February 18, 2010 MATH 682 Notes Combinatorics and Graph Theory II Proof. If G is acyclic, it has a single face with boundary demonstrably of length 2 | G | > 3. Oth- erwise, every face has a nontrivial walk on its boundary, which could be reduced to a cycle by ignoring the edges which jut into the face. Since cycles have length of at least 3, the boundary walks have length of at least 3....
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