Note on robust critical graphs with large odd girth

A graph G is (k+1)-critical if it is not k-colourable but G−e is k-colourable for any edge e ∈ E(G). In this paper we show that for any integers k ≥ 3 and l ≥ 5 there exists a constant c = c(k, l) > 0, such that for all ñ, there exists a (k + 1)-critical graph G on n vertices with n > ñ and odd girth at least l, which can be made (k − 1)-colourable only by the omission of at least cn2 edges.