Locally 2‐arc transitive graphs, homogeneous factorizations, and partial linear spaces

Abstract

We establish natural bijections between three different classes of combinatorial objects; namely certain families of locally 2-arc transitive graphs, partial linear spaces, and homogeneous factorizations of arc-transitive graphs. Moreover, the bijections intertwine the actions of the relevant automorphism groups. Thus constructions in any of these areas provide examples for the others.

Arc‐transitive homogeneous factorizations and affine planes

“…The case where n ¼ 81 is particularly interesting since besides giving rise to Hamming graphs as arc-transitive homogeneous factors, it also give rise to the exceptional nearfield plane of order 9 studied by Kantor in [13]. Recently, Giudici, Li, and Praeger [9] studied the interaction between partial linear spaces, homogeneous factorizations, and locally 2-arc-transitive graphs. However, our example in Theorem 1.4 does not fit the general framework of [9], and that was another motivation for our further analysis.…”

“…Recently, Giudici, Li, and Praeger [9] studied the interaction between partial linear spaces, homogeneous factorizations, and locally 2-arc-transitive graphs. However, our example in Theorem 1.4 does not fit the general framework of [9], and that was another motivation for our further analysis.…”

Countable locally2-arc-transitive bipartite graphs

“…The locally s-arc-transitive graphs have received a lot of attention in the literature partly due to their links with areas of mathematics such as generalized n-gons, groups with a (B, N )-pair of rank two, Moufang polygons and Tutte's m-cages; see [32] and [7] for more details. Important recent papers about local s-arc-transitivity include [15,16,17,18]. In particular in [15] a programme of study of locally sarc-transitive graphs for which G acts intransitively on vertices was initiated, with the O'Nan-Scott Theorem for quasi-primitive groups playing an important role in their reduction theorem.…”

“…In particular in [15] a programme of study of locally sarc-transitive graphs for which G acts intransitively on vertices was initiated, with the O'Nan-Scott Theorem for quasi-primitive groups playing an important role in their reduction theorem. Interesting connections with semilinear spaces and homogeneous factorizations were explored in [17].…”

Locally s -distance transitive graphs and pairwise transitive designs

“…Every locally s-arc transitive graph is locally s-distance transitive but the converse does not hold. Connections between starlike locally s-arc transitive graphs, partial linear spaces and homogeneous factorisations were explored in [10], while the basic starlike locally (G, s)-arc transitive graphs outlined in the program initiated in [8] were investigated in [9].…”