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 \(2s-1\) $2s-1$ or \(2s-2\) $2s-2$ , where \(s=4,5,6,7,8\) $s=4,5,6,7,8$ . In this paper, we study the \(s=4\) $s=4$ 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 \(\Gamma \) $\mathrm{\Gamma }$ is a cover of its quotient \(\Gamma _N\) ${\mathrm{\Gamma }}_N$ modulo the N-orbits and either \(\Gamma _N\) ${\mathrm{\Gamma }}_N$ is (G/N, s)-geodesic-transitive where \(s=\min \{4,\textrm{diam}(\Gamma _N)\}\ge 3\) $s=min\{4,\text{diam}({\mathrm{\Gamma }}_N)\}\ge 3$ , or \(\Gamma _N\) ${\mathrm{\Gamma }}_N$ 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 \(\Gamma \) $\mathrm{\Gamma }$ when \(\Gamma _N\) ${\mathrm{\Gamma }}_N$ is strongly regular.