The $k$-Colouring problem is to decide if a graph $G$ has a $K$-colouring . We prove for certain small values of $s$ that $3$ ispolynomial-time solvable for $C_s$-free graphs of diameter $2$ and $(C_4,C__s)$) We complement these results with somehardness result for diameter $4$ . In previous work (MFCS 2019) we examined the effect of bounding the diameter on the complexity of the problem . In fact, our results hold for the moregeneral problem List $3-Colourouring. In fact

Author(s) : Barnaby Martin, Daniel Paulusma, Siani Smith

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Keywords : diameter - c - problem - colouring - results -

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