Tutorial 5. - University of St Andrews

UNIVERSITY OF ST ANDREWS
School of Mathematics and Statistics
MT4514 Graph Theory: Tutorial 5.
1. (i) Prove that any graph in which all the vertices have degree ≤ 3 is k-
colourable with k ≤ 4.
(ii)
Give an example of a map on the plane in which no subset of four regions
all touch one another but which is not 3-colourable. (The graph of such
a map will not contain K 4 as a subgraph.)
2. Prove that any connected planar graph is the graph of some map.
3. Let H be the graph obtained by deleting m ends (vertices of degree one)
from G. Prove that
P G (k) = (k − 1) m P H (k).
4. (i) Prove that the chromatic polynomial of any tree (a connected graph
with no circuits) with n vertices is k(k − 1) n−1 .
(ii)
Prove that the chromatic polynomial of the complete graph K n is
k!/(k − n)!.
5. (i) If C n is the chromatic polynomial of the n-circuit (a connected graph
with n vertices of degree 2), prove that C n satisfies C n = k(k − 1) n−1 −
C n−1
(ii)
Hence prove that the chromatic polynomial of the n-circuit is given by
(k − 1)((k − 1) n−1 + (−1) n ).
(iii)
Find the chromatic number of the n-circuit.
6. Suppose that ABC is a triangle in the graph G and that the degree of B is
2. If H is the graph obtained by deleting the vertex B (and the edges AB
and BC) then prove that P G (k) = (k − 2)P H (k).
7. Find the chromatic polynomials of the following graphs and hence determine
their chromatic numbers, and find a colouring with that many colours.

8. Let G be a connected planar graph such that each region into which G divides
the plane is bounded by a circuit of at least three edges. Show that:
(i)
if the faces of G can be 2-coloured then G is Eulerian
(ii) if the faces of G can be 3-coloured and every vertex of G has degree 3
then every face of G is bounded by an even number of edges.
9. Use induction and the deletion-contraction property to show that the chromatic
polynomial of a simple graph G with v vertices and e edges has the
form
P G (k) = k v − ek v−1 + lower order terms
with the signs of the terms alternately + and − .