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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)|.
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