Brooks' Vertex-Colouring Theorem in Linear Time

Brooks' Theorem [R. L. Brooks, On Colouring the Nodes of a Network, Proc. Cambridge Philos. Soc.} 37:194-197, 1941] states that every graph $G$ with maximum degree $Δ$, has a vertex-colouring with $Δ$ colours, unless $G$ is a complete graph or an odd cycle, in which case $Δ+1$ colours are required. Lovász [L. Lovász, Three short proofs in graph theory, J. Combin. Theory Ser. 19:269-271, 1975] gives an algorithmic proof of Brooks' Theorem. Unfortunately this proof is missing important details and it is thus unclear whether it leads to a linear time algorithm. In this paper we give a complete description of the proof of Lovász, and we derive a linear time algorithm for determining the vertex-colouring guaranteed by Brooks' Theorem.

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