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 over(n, ̃), there exists a (k + 1)-critical graph G on n vertices with n > over(n, ̃) and odd girth at least ℓ, which can be made (k - 1)-colourable only by the omission of at least c n² edges.