Graph colouring via the discharging method

• Authors: Michael Molloy;Mohammad Reza Salavatipour
• Venue: Graph colouring via the discharging method
• Year: 2003

In this thesis we study two colouring problems on planar graphs. The main technique we use is the Discharging Method, which was used to prove the Four Colour Theorem. The first problem we study is a conjecture of Steinberg which states that every planar graph without 4 and 5-cycles is 3-colourable. Erdös relaxed this conjecture by asking if there exists a k such that every planar graph without cycles of size in {4,…,k} is 3-colourable. Abbott and Zhou [1] answered the question of Erdös by showing that such a k exists and can be as small as 11, i.e. any planar graph without cycles of size in {4,…,11} is 3-colourable. This result was improve by Borodin [15] to k = 10, and by Borodin [14] and by Sanders and Zhao [49] to k = 9. We improve these results by two steps. First we reduce k down to 8. That is, we show every planar graph without cycles of size in {4,…,8} is 3-colourable. This theorem is constructive and yields an O(n 2) time algorithm for 3-colouring such graphs. Then we improve this result one step further, by showing that every planar graph without cycles of size in {4,…,7} is 3-colourable. This theorem too is constructive and yields an O(n 3) time 3-colouring algorithm for such graphs. The second problem is the problem of colouring the squares of planar graphs. Equivalently, it is the problem of colouring the vertices of a planar graph in such a way that vertices at distance at most 2 from each other get different colours. This is also known as distance-2-colouring. Wegner in 1977 conjectured that, for every planar graph G with maximum degree D ≥ 8, the minimum number of colours required in any distance-2-colouring of G is at most 32D + 1. This conjecture, if true, would be the best possible upper bound for the number of colours needed, in terms of D . The previously best known bound for this quantity is 95D + 1, for graphs with D ≥ 47, by Borodin et al. [16, 17]. We improve this result by showing that 53D + O(1) colours are enough for a distance-2-colouring of a planar graph with maximum degree D . We also provide a better bound for large values of D . Then we generalize this result to L(p, q )-labelings of planar graphs. An L(p, q )-labeling of a graph G is an assignment of integers from {0,…,k} to the vertices of G such that every two adjacent vertices in G receive integers that are at least p apart and every two vertices at distance two from each other receive integers that are at least q apart. The minimum k for which there is an L( p, q)-labeling of G is denoted by lpqG . We prove that for any planar graph G: lpqG ≤ q 53D + O(p + q). This improves the previously known bound of (4q − 2) D + O(p + q), by Van den Huevel and McGuinness [57]. All these results are constructive; we provide efficient algorithms for distance-2-colouring of planar graphs with at most 53D + O(1) colours and for L( p, q)-labeling of planar graphs using only q53D + O(p + q) colours.