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 n ̃ , there exists a ( k + 1 ) -critical graph G on n vertices with n > n ̃ and odd girth at least ℓ , which can be made ( k − 1 ) -colourable only by the omission of at least c n 2 edges.