Let G = (V,E) be a simple undirected graph of n vertices.
(a) Suppose that G has an even number n of vertices, of which n/2 have maximum degree n - 1, and the remaining n/2 have degree 2.
Show that n must be be a multiple of 4 (5 marks).
(b) Suppose that, for some integer m, and for every k ∈ {1,...,m}, the graph G has exactly k vertices of degree m + 1 - k (that is, m vertices of degree 1, m - 1 vertices of degree 2, m - 2 vertices of degree 3, and so on).
Show that, if m is odd, then m + 1 must be a multiple of 4 (5 marks).
(c) Suppose that G has an odd number of vertices n ≥ 3 and that no vertex has degree larger than (n - 3)/2. Show that there must exist three vertices with the same degree (10 marks).