Arc‐transitive homogeneous factorizations and affine planes

Abstract

We study arc-transitive homogeneous factorizations of a complete graph on 81 vertices and give an almost complete description of all possible factors that arise. In particular, we show that there exists an arc-transitive homogeneous factorization of K 81 such that the factors are isomorphic to the Hamming graph Hð9; 2Þ. This gives rise to an edge partition of K 81 into 90 copies of K 9 and it turns out that these copies of K 9 form the blocks of the exceptional nearfield affine plane of order 9.

Homogeneous factorisations of complete graphs with edge-transitive factors

“…(e) The exceptional example in Part 2(d) is associated with the exceptional affine 2-transitive group G with G 0 ≤ 2 1+4 .S 5 , and M 0 = 2 1+4 ; it is explored in detail in [23] and [24].…”

“…It follows from this lemma that there is only one edge partition arising in this case, namely E(M) for the unique group M = T E. The factors i of this factorisation were identified in [23] as Hamming graphs H (9, 2). Thus we have the following proposition, which completes the proof of Theorem 1.1.…”

Homogeneously almost self-complementary graphs

“…The latter, however, has a unique index-2 subgroup, ruling it out as a candidate for G 0 . This leaves us with the case p r = 3 4 , which has been studied by Lim in [6]. From his result [6, Theorem 1.2] it follows that in this case, arc-transitive homogeneous factorizations of index 4 with G 0 ≤ ΓL 1 (F) do not exist.…”

“…This result is used in Subsection 3.2 to obtain an analogous classification of arc-transitive symmetric index-4 homogeneous factorizations (M, G, K n , P) with G/M ∼ = Z 2 × Z 2 . Our classification will also rely on a result by Lim, who discussed arc-transitive homogeneous factorizations of complete graphs in [6,7].…”

Brick assignments and homogeneously almost self-complementary graphs

“…This result is used in Section 3.2 to obtain an analogous classification of arc-transitive symmetric index-4 homogeneous factorizations (M, G, K n , P) with G/M ∼ = Z 2 × Z 2 . Our classification will also rely on a result by Lim, who discussed arc-transitive homogeneous factorizations of complete graphs in [8,9].…”