Distance 4 curves on closed surfaces of arbitrary genus

Let S-g denote a closed, orientable surface of genus g >= 2 and C(S-g) be the associated curve graph. Let curves alpha, beta and the Dehn twist of alpha about beta represent vertices a, b and c in C(S-g), respectively. If a and b are at a distance 3 from each other then we show that c is at a distance 4 from a and 3 from b. This produces many tractable examples of distance 4 vertices in C(S-g). As an application we show that the minimum intersection number of any two curves at a distance 4 on S-g is at most (2g - 1)(2). (C) 2022 Elsevier B.V. All rights reserved.