Note on robust critical graphs with large odd girth

A graph GG is (k+1)(k+1)-critical if it is not kk-colourable but G−eG−e is kk-colourable for any edge e∈E(G)e∈E(G). In this paper we show that for any integers k≥3k≥3 and l≥5l≥5 there exists a constant c=c(k,l)>0c=c(k,l)>0, such that for all ñ, there exists a (k+1)(k+1)-critical graph GG on nn vertices with n>ñ and odd girth at least ℓℓ, which can be made (k−1)(k−1)-colourable only by the omission of at least cn2cn2 edges.