Mathematics and Statistics, Part A: Graph Theory Revision Exercises 1. Prove Caley’s theorem. There are nn−2 distinct labelled trees on n vertices. (i) Sketch a tree with 11 edges and find the Pr¨ ufer code corresponding to it. (ii) Explain how to simulate random trees with n vertices such that each labelled tree is equally likely. (iii) Find the probability distribution of the number of leaves in a random tree generated in (ii). 2. Let G be a forest of k trees with a total of n vertices. Show that G has n − k edges. 3. Prove Euler’s formula. Let G be a plane drawing of a connected planar graph, and let m, n and f denote the number of vertices, edges, and faces of G. Then n − m + f = 2. 4. Prove that K5 and K3,3 are non-planar. (i) Is K4 planar? (ii) Is K3,2 planar? (iii) Is K6 planar? 5. Let G be a planar graph. Explain the construction of a dual graph G∗ . Show that n∗ = f, m∗ = m and f ∗ = n. 6. If G is a plane connected graph show that G∗∗ is isomrphic to G. Show that this is not true if G is not connected by an example. 7. Prove that a map G is 2-colourable if and only if G is an Eulerian graph. 8. Qk is a k-cube. (i) How many vertices does Qk have? (ii) How many edges does Qk have? (iii) Show that Q3 is a Hamiltonian graph. (iv) Find a vertex colouring of Q3 with a minimal number of colours.

(v) Show that the vertices of Q3 can be coloured to make it a bipartite graph. (vi) What is the minimum number of colours needed to colour Q2k+1 . 9. Use Fleury’s algorithm to obtain an Eulerian trail in the following graph, starting with the edges uv, vz.

10.

(i) Explain Hall’s marriage problem. (ii) Prove that if there are m girls then a necessary and sufficient condition for the solution of the marriage problem is that each set of k girls collectively knows at least k boys, for 1 ≤ k ≤ m. (iii) Explain why the graph below has no complete matching from V1 to V2 . When does the marriage condition fail?

(iv) Let S1 , S2 , . . . , Sm be subsets of a non-empty set E. The subsets are not necessarily distinct. Deduce a necessary and sufficient condition that there exists distinct elements e1 , e2 , . . . , em such that ei ∈ Si , i = 1, . . . , m. It is helpful to consider a bipartite graph G(V1 , V2 ), where V1 = {1, 2, . . . , m} and V2 = E with an edge set {(i, ej ) : ej ∈ Si }.

Graph Theory Exam Questions 2004 Short question. 11. Euler’s theorem (1750). Let G be a plane drawing of a connected planar graph, and let n, m, f denote the number of vertices, edges, and faces of G. Then n − m + f = 2. Euler’s formula can be assumed in the question. (i) If G is a connected simple planar graph with n ≥ 3 vertices and m edges show that 3f ≤ 2m, and from Euler’s theorem m ≤ 3n − 6. (ii) (a) Show that the complete graph with five vertices K5 is nonplanar. (b) Find the minimum number of edges that have to be removed from K5 to make the reduced graph planar. Sketch the reduced graph as a planar graph. (iii) Find the chromatic number of K5 and the chromatic number of the reduced graph in (ii)(b). Long question. 12. A theorem of Caley (1889) states that there are nn−2 distinct labelled trees on n vertices. (i) Prove Caley’s theorem by showing that there is a 1-1 correspondence between the set of labelled trees of order n and the set of sequences (a1 , a2 , . . . , an−2 ), where each ai is an integer satisfying 1 ≤ ai ≤ n. The encoding of a tree in this way is called the Pr¨ ufer code. (ii) Find the Pr¨ ufer code corresponding to the tree below.

(iii) Find the tree corresponding to the Pr¨ ufer code (3, 1, 1, 2, 1). (iv) Let ρ(i) be the degree of the vertex labelled i in a tree with n vertices labelled 1, 2, . . . , n. (a) Explain how to read the degree of vertex labelled i from the Pr¨ ufer code of a tree. (b) Show that the number of labelled trees with vertex degrees ρ(1), . . . , ρ(n) is (n − 2)! (ρ(1) − 1)! · · · (ρ(n) − 1)! Graph Theory Exam Questions 2005 Short question. 13.

(i) Define an Eulerian graph, and state Euler’s theorem which characterizes such graphs in terms of their vertex degree. (ii) Explain the concept of a map G. (iii) Prove that a map G is 2-colourable if and only if G is an Eulerian graph. (iv) Let G∗ be the dual of an Eulerian graph G. What are the implications of (iii) in G∗ ? (v) Sketch the cube in three dimensions, Q3 , and find a 2-colour vertex colouring of the cube.

Long question. 14. Hall’s (1935) marriage problem is the following. If there is a finite set of girls, each of whom knows several boys, under what conditions can all the girls marry the boys in such a way that each girl marries a boy she knows? (i) Explain the correspondence between the marriage problem and a bipartite graph. (ii) Prove Hall’s theorem which states that a necessary and sufficient condition for a solution of the marriage problem is that each set of k girls collectively knows at least k boys, for 1 ≤ k ≤ m. In your answer define a complete matching from V1 to V2 in a bipartite graph G(V1 , V2 ). (iii) Three girls a, b, c know four boys w, x, y, z such that a knows w, y, z; b knows x, z; and c knows x, y. (a) Draw the bipartite graph according to this table of relationships. (b) Find five different solutions to the marriage problem. (c) Check the marriage condition for this problem. (iv) In a random bipartite graph G(V1 , V2 ) with |V1 | = α, |V2 | = β ≥ α a fixed number of distinct edges α join vertices from V1 to V2 . The α edges are chosen randomly from the edge set V1 × V2 . Show that the probability of a complete matching from V1 to V2 is
β! α(β − 1) ! (αβ)!(β − α)!