Finite 4-geodesic-transitive graphs with bounded girth

Praeger and the first author in Jin and Praeger (J Combin Theory Ser A 178:105349, 2021) asked the following problem: classify s-geodesic-transitive graphs of girth \(2s-1\) or \(2s-2\), where \(s=4,5,6,7,8\). In this paper, we study the \(s=4\) case, that is, study the family of finite (G, 4)-geodesic-transitive graphs of girth 6 or 7 for some group G of automorphisms. A reduction result on this family of graphs is first given. Let N be a normal subgroup of G which has at least 3 orbits on the vertex set. We show that such a graph \(\Gamma \) is a cover of its quotient \(\Gamma _N\) modulo the N-orbits and either \(\Gamma _N\) is (G/N, s)-geodesic-transitive where \(s=\min \{4,\textrm{diam}(\Gamma _N)\}\ge 3\), or \(\Gamma _N\) is a (G/N, 2)-arc-transitive strongly regular graph. Next, using the classification of 2-arc-transitive strongly regular graphs, we determine all the (G, 4)-geodesic-transitive covers \(\Gamma \) when \(\Gamma _N\) is strongly regular.