A Short Proof of the Hajnal–Szemerédi Theorem on Equitable Colouring

A proper vertex colouring of a graph is equitable if the sizes of colour classes differ by at most one. We present a new shorter proof of the celebrated Hajnal–Szemerédi theorem: for every positive integer r , every graph with maximum degree at most r has an equitable colouring with r +1 colours. The proof yields a polynomial time algorithm for such colourings.