Homework 7 Solutions
Outerplanar A graph G is outerplanar if it can be drawn in the plane so that all vertices lie on the infinite face. 1. Show that G is outerplanar if and only if G has no K2,3 or K4 minor. Solution: Let G+ be the graph obtained from G
The Probabilistic Method I
In this section, graphs are assumed to have no loops or parallel edges. Theorem 17.1 (Erds) If o t 3. Proof: Construct a random red/blue colouring of the edges of Kn by colouring each edge
1 independently either red or blue with
Graph Colouring IV: edge colouring
In this section, we shall assume that all graphs are loopless, but we permit parallel edges. Edge Colouring A k-edge colouring of a graph G is a map g : E(G) S where |S| = k. The elements of S are called colours, and the
Graph Colouring I: upper and lower bounds
In this section, we shall assume that all graphs are simple and loopless. Colouring: A k-colouring of a graph G is a map f : V (G) S where |S| = k. The elements of S are called colours, and the vertices of one col
Higher Surfaces I: embeddings and the torus
In this section, graphs are permitted to have loops and parallel edges. Surface: A surface is a topological space which appears locally like the plane. More precisely, for every point x in the surface, there is
Graph Colouring III: counting colourings
In this section graphs may have loops and parallel edges. Counting Proper Colourings For any graph G and any positive integer t, we let (G; t) denote the number of proper t-colourings f : V (G) cfw_1, 2, . . . , t
Graph Embedding I: drawing & duality
In this section, graphs are permitted to have loops and multiple edges. Drawing If G is a graph, a drawing of G in the plane is a function f which assigns each vertex of G a distinct point in the plane, and assigns eac
Graph Embedding II: Euler's Formula
In this section, graphs are permitted to have loops and multiple edges. Theorem 6.1 (Euler's Formula) If G is a connected plane graph with v vertices, e edges, and f faces, then v - e + f = 2. Proof: We proceed by induc
Graph Colouring II: structure
In this section, we shall assume that all graphs are simple and loopless. Theorem 2.1 (Gallai-Roy-Vitaver) If D is an orientation of G and the longest directed path in D has length t, then (G) t + 1. Furthermore, equality hol
Extremal Graph Theory I: Turn & Ramsey a
In this section, graphs are assumed to have no loops or parallel edges. Complete t-Partite If m1 , m2 , . . . , mt are positive integers, the complete t-partite graph Km1 ,m2 ,.,mt is a simple graph with vertex par
The Probabilistic Method II
In this section, graphs are assumed to have no loops or parallel edges. Theorem 18.1 (Erds) For every g, k there exists a graph with chromatic number k and o no cycle of length g.
1 Proof: For typographical reasons, we set = 2g
Hamiltonian Cycles I: the basics
In this section, graphs are assumed to have no loops or parallel edges. Hamiltonian: A cycle C of a graph G is Hamiltonian if V (C) = V (G). If G has such a cycle, we say that G is Hamiltonian. A path P G for which V (P )
Homework 8 Solutions
1. Let G be a 3-connected graph with crossing number > 0 and assume that G e is planar for every edge e. Show that G is either K5 or K3,3 . Solution: Since G is not planar, and G is 3-connected, it follows from last weeks homework tha
Homework 9 Solutions
1. A mouse eats its way through a 3 3 3 cube of cheese by eating the 1 1 1 subcubes, one after another, in such a way that consecutively consumed subcubes share a face. If the mouse starts at a corner subcube, can it end at the middle
Homework 4
1. Use Tutte's 1-factor Theorem to prove that every connected line graph of even order has a perfect matching. Conclude from this that the edges of a simple connected graph of even size can be partitioned into paths of length 2. Solution: Let G
Homework 6 Solutions
1. Prove that a set of edges in a connected plane graph G form a spanning tree if and only if the duals of the remaining edges form a spanning tree in G . Solution see problem 3 from homework 5. 2. Use Exercise 1 to obtain a new proof
Homework 5 Solutions
1. If G is a 2-connected simple plane graph with minimum degree 3, does it follow that the dual graph G is simple? Give a proof or a counterexample. Solution: The following graph is a counterexample:
2. Prove that contracting an edge
Homework 1 - Solutions
1. Show that (G) |V (G)| (G) + 1. Solution: Choose an independent set in G of size (G), and give all vertices in this set colour 1. Then assign a distinct colour to each remaining vertex. This gives a proper colouring of G which use
Hamiltonian Cycles II: planarity
In this section, graphs are assumed to have no loops or parallel edges. Observation 14.1 Every 3-regular graph which is Hamiltonian is 3-edge-colourable. Proof: If G is a 3-regular graph with Hamiltonian cycle C, then we m
Extremal Graph Theory II: more Ramsey Theory
In this section, graphs are assumed to have no loops or parallel edges. Hypergraph: A hypergraph H consists of a set of vertices, denoted V (H), a set of edges (sometimes called hyperedges), denoted E(H), and a
Higher Surfaces II: genus
In this section, graphs are permitted to have loops and parallel edges. Handles: To add a handle to a surface S, we remove two disjoint discs from it, and then add a cylinder, so that each end of the cylinder is identied with the