For any integer k, it is deduced that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G)≥log∣G∣/13logk and maximum degree Δ( G)≤6k13 can exist.Expand

Counterexamples to the conjecture of Xu that every uniquely r-colorable graph of order n with exactly (r?1)n?(r2) edges must contain a Kr. Combin. Theory Ser. B50 conjecture are constructed.Expand

Tutte [1], writing under a pseudonym, was the first to prove that a graph with a large chromatic number need not contain a triangle. The result was rediscovered by Zykov [5] and Mycielski [4]. Erdös… Expand