Hints for Chapter 8
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Hints for Chapter 8 1.
For the induction step, partition the vertex set of the given graph G into two sets V1 and V2 so that colourings of G [ V1 ] and G [ V2 ] can be combined to a colouring of G.
2. −
3.
Imitate the start of the proof of Lemma 8.1.3. Does a large chromatic number force up the average degree? If in doubt, consult Chapter 5.
4.+ Try parallel paths in the grid as branch sets. 5.+ How can we best make a T K 2r fit into a Ks,s when we want to keep s small? 6.
Split the argument into the cases of k = 0 and k > 1.
7.
How are the two lemmas used in the proof of the theorem?
8.
Study the motivational chat preceding the definition of f in the proof.
9.+ Consider your favourite graphs with high average degree and low chromatic number. Which trees do they contain induced? Is there some reason to expect that exactly these trees may always be found induced in graphs of large average degree and small chromatic number? 10.− What does planarity have to do with minors? 11.− Consider a suitable supergraph. 12.− Average degree. 13.+ Show by induction on |G| that any 3-colouring of an induced cycle in G 6< K 4 extends to all of G. 14.+ Reduce the statement to critical k-chromatic graphs and apply Vizing’s theorem. 15.
(i) is easy. In the first part of (ii), distinguish between the cases that the graph is or is not separated by a K χ(G)−1 . Show the second part by induction on the chromatic number. In the induction step split the vertex set of the graph into two subsets.
16.
Induction on the number of construction steps.
17.
Induction on |G|.
18.
Note the previous exercise.
19.
Which of the graphs constructed as in Theorem 8.3.4 have the largest average degree?
20.
Which of the graphs constructed as in the hint have the largest average degree?
21.
Consider the subgraph of G induced by the neighbours of x.