Equitable Colourings Of D-Degenerate Graphs

In many applications of graph colouring the sizes of colour classes should not be too large. For example, in scheduling jobs (some of which could be performed at the same time), it is not good if the resulting schedule requires many jobs to occur at some specific time. An application of this type is discussed in [8]. A possible formalization of this restriction is the notion of equitable colouring. A proper vertex colouring of a graph is called equitable if the sizes of colour classes differ by at most 1. A graph may have an equitable k -colouring (i.e., an equitable colouring with k colours) but have no equitable (k + 1)-colouring. For example, the complete bipartite graph K7,7 has an equitable k -colouring for k = 2, 4, 6 and 8, but has no equitable k -colouring for k = 3, 5 and 7. Thus, it is natural to look for the minimum number, eq(G), such that, for every k eq(G), G has an equitable k -colouring. A good survey on equitable colourings of graphs is given in [5]. Hajnal and Szemerédi [3], answering a question of Erdó́s, proved that, for every graph G , eq(G) ∆(G) + 1. Recently, Pemmaraju [7] used equitable colourings to give new bounds on the tail of the distribution of the sum of random variables. He applied different theorems on equitable colourings for different situations. If the dependence graph of variables had a bounded maximum degree, he applied the above-mentioned Hajnal–Szemerédi theorem [3]; for trees he used a bound of Bollobás