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(1) (2) (3) (4) Solution Sketch Homework Nine Part II (of II) Show that Q+ is countable and use this fact to show that Q is countable. Solution: See pages 247 − 250 in the textbook. A cycle in a graph is a path whose initial and terminal vertices coincide. A graph with no cycles is called a forest. Show that if a forest has v vertices, e edges and c connected components, then v = e + c. Think about why this is surprising and weird. Solution: See 220-202 website, Homework 8 solutions. A graph is called bipartite if its set of vertices can be separated into two disjoint sets U and V such that every edge in the graph connects a vertex in U to a vertex in V . Prove that a finite graph is bipartite if and only if it contains no cycles of odd length. Solution: See 220-202 website, Homework 8 solutions. A colouring of a graph G is a function θ : V (G) → C from V (G) to a finite set of colours C := {c1 , c2 , c3 , . . . , ck } such that no two adjacent vertices are assigned the same colour. If G can be coloured with a set of k colours, then we say G can be kcoloured. The minimal value of k for which G can be k-coloured is the chromatic number of G and is denoted by χ(G). (a) Find χ(G) if G is a cycle with at least three vertices. Solution: Algorithmically, if |V | is even, χ = 2. If |V | is odd, χ = 3. (b) Find χ(G) if G is a complete graph. Solution: Here, χ = |V | because all vertices are adjacent. (c) Find χ(G) if G is a tree with at least one edge. Solution: Here, χ = 2 because trees are bipartite. (d) Find χ(G) if G is bipartite with at least one edge. Solution: Here, χ = 2 because vertices can be partitioned into two independent sets. (e) Find χ(G) for your favourite graph G. (f) Describe a “real world” problem that can be modelled by a graph G where knowing χ(G) is useful. Hint. As usual, it may be useful to draw several small examples before describing the general cases. Your answers may be given as a function of |V (G)| or as a function of the parity of |V (G)|. 1