Show that an r-regular non-bipartite graph does not necessarily admit a 1-factorization, even if it has an even number of vertices. Hint: Consider the Petersen graph. Let G be a graph on 3n vertices. A 2-factor of G is called a triangle factor or a -factor if it is the disjoint union of n cycles of length 3. Show that it is possible to decompose the graph K6n into one -factor and 6n − 3 1-factors. Hint: View the vertex set as the union of three sets R, S, T of cardinality 2n each, and consider regular bipartite graphs on all pairs of these sets. Furthermore, use Exercise 1.1.2.