MT4514: Graph Theory

36 Colva M. Roney-DougalHere is an edge colouring of K 6 using the method of Theorem 2.4.3. Factorisation of graphsWhen considering Hall’s Marriage Theorem we were interested in bipartite graphs.In this section we consider sets of independent edges of graphs that need not bebipartite.Definition 3.1. A factor of a graph G is a subgraph of G which uses all of thevertices of G and has at least one edge. We say that G can be factorised if it is theunion of factors G 1 , . . . , G k , and they have no edges in common. An n-factor of Gis a factor of G that is regular of degree n.Example 3.2. Here is a graph and a 1-factorisation.Lemma 3.3. If G has a 1-factor H then |V (G)| is even.Proof. Every vertex in H has degree 1, so H is a set of independent edges. ThusH has an even number of vertices. Since H uses all vertices of G, the graph G mustalso have an even number of vertices. Theorem 3.4.1. The complete graph K 2n+1 is not 1-factorable.2. The complete graph K 2n is 1-factorable.Proof. 1. It is an immediate consequence of Lemma 3.3 that K 2n+1 does nothave any 1-factors, and so is not 1-factorable.2. To prove this we need to divide up the edges of K 2n into (2n − 1) 1-factors, eachcontaining n edges, since K 2n has 2n vertices and n(2n − 1) edges.Let the vertices of K 2n be {v 0 , . . . , v 2n−1 }. For 0 ≤ i ≤ 2n − 2 we define a setof edges:X i = {v i v 2n−1 } ∪ {v i−j v i+j : 1 ≤ j ≤ n − 1}where the subscripts are taken modulo 2n − 1.It is clear that X i contains n edges, and that no vertex occurs more than oncein X i , so that ({v 0 , . . . , v 2n−1 }, X i ) is a 1-factor of K n .