Graphs, colourings and the four-colour theorem

Corrections

• p. 29: in line 1, 'negative' should read 'non-positive', and in Lemma 3.17 and Proposition 3.18 the summation should start with i=0.
• p. 38: the proof of Theorem 4.11 is incomplete, as it does not deal with the case when the collapsing produces a multigraph which is not a graph.
• p. 57: in Exercise 5.10, Delta = Delta(G).
• p. 61: the proof of Corollary 6.2 is inaccurate, as the case p-q+r=1 may also occur. It is better to continue adding edges until the faces are 2-cells, when p-q+r=0.
• p. 120: the Kempe-chain arguments in the proof of Theorem 10.9 are wrong. In case 4, either there is a blue-green chain from v₂ to v₇, in which case we can change v₁ to yellow; or there is a blue-green chain from v₅ to v₇, in which case we can change v₆ to yellow; or there is neither, in which case we can change v₇ to green. In all three cases, we can now complete the colouring. Similarly, in case 5, either there is a blue-yellow chain from v₂ to v₅, in which case we can change v₃ to red and v₄ to green; or there is a blue-yellow chain from v ₅ to v₇, in which case we can change v₆ to red, and we are back in case 3; or there is neither, in which case we can change v₅ to yellow. In all cases, we can now complete the colouring.

My thanks to Marijke van Gans and Peter Kratz for notifying me of errors.