Let G be a simple graph that has 10 vertices and 38 edges. Prove that G contains K4 (the complete graph on four vertices) as an induced subgraph. (6) Let G be a simple graph in which all vertices have degree four. Prove that it is possible to color the edges of G orange or blue so that each vertex is adjacent to two orange edges and two blue edges. (7) How many different simple graphs are there on the vertex set [n]? (8) An automorphism of a graph G is an isomorphism between G and G itself. That is, the permutation f of the vertex set of G is an automorphism of G if for any two vertices x and y of G, the number of edges between x and y is equal to the number of edges between f(x) and f(y). How many automorphisms do the following (labeled) graphs have? (a) The complete graph Kn on n vertices. (b) The cycle Cn on n vertices. (c) The path Pn on n vertices. (d) The star Sn on n vertices. (This graph has one vertex of degree n - 1, and n - 1 vertices of degree 1.) (9) Prove that there are more than 6600 pairwise non-isomorphic graphs on eight labeled vertices. (10) Is it true that the number of people currently living on our planet and having an odd number of siblings is even? (11) Is it true that (a) if a simple graph has a closed Eulerian trail, then it has a Hamiltonian cycle? (b) if a simple graph has a Hamiltonian cycle, then it has a closed Eulerian trail?