On primitive permutation groups with small suborbits and their orbital graphs

In this paper, we study finite primitive permutation groups with a small suborbit. Based on the classification result of Quirin [Math. Z. 122 (1971) 267] and Wang [Comm. Algebra 20 (1992) 889], we first produce a precise list of primitive permutation groups with a suborbit of length 4. In particular, we show that there exist no examples of such groups with the point stabiliser of order 2436, clarifying an uncertain question (since 1970s). Then we analyse the orbital graphs of primitive permutation groups with a suborbit of length 3 or of length 4. We obtain a complete classification of vertex-primitive arc-transitive graphs of valency 3 and valency 4, and we prove that there exist no vertex-primitive half-arc-transitive graphs of valency less than 10. Finally, we construct vertex-primitive half-arc-transitive graphs of valency 2k for infinitely many integers k, with 14 as the smallest valency.