By Joseph Muscat
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2 to derive bounds on the number of edges of planar graphs. We need two more definitions. An edge e of a connected graph G is called a bridge if G \ e is not connected. The girth of a graph containing cycles is the length of a shortest cycle. 3. Let G be a connected planar graph on n vertices. If G is acyclic, then G has precisely n − 1 edges. If G has girth at least g, then G can have at most g(n−2) g−2 edges. Proof. 8. Thus let G be a connected planar graph having n vertices, m edges and girth at least g.
1 (Euler’s theorem). Let G be a connected multigraph. Then the following statements are equivalent: (a) G is Eulerian. (b) Each vertex of G has even degree. (c) The edge set of G can be partitioned into cycles. Proof: We first assume that G is Eulerian and pick an Euler tour, say C. Each occurrence of a vertex v in C adds 2 to its degree. As each edge of G occurs exactly once in C, all vertices must have even degree. The reader should work out this argument in detail. 3 4 Some authors denote the structure we call a multigraph by graph; graphs according to our definition are then called simple graphs.
Then 1 has the form 1 = k − i for some odd i, so that 1 has an away game on that day. Similarly it can be shown that the vertex complementary to 2i (for i = 1, . . , n − 1) is the vertex 2i + 1. Now we still have the problem of finding a schedule for the return round of the league. Choose oriented factorizations DH and DR for the first and second round. Of course, we want D = DH ∪ DR to be a complete orientation of K2n ; hence ji should occur as an edge in DR if ij occurs in DH . If this is the case, D is called a league schedule for 2n teams.