On partitioning the orbitals of a transitive permutation group

Abstract

Abstract. Let G be a permutation group on a set Ω with a transitive normal subgroup M . Then G acts on the set Orbl(M, Ω) of nontrivial M -orbitals in the natural way, and here we are interested in the case where Orbl(M, Ω) has a partition P such that G acts transitively on P. The problem of characterising such tuples (M, G, Ω, P), called TODs, arises naturally in permutation group theory, and also occurs in number theory and combinatorics. The case where |P| is a prime-power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials. The case where |P| = 2 exactly corresponds to self-complementary vertex-transitive graphs, while the general case corresponds to a type of isomorphic factorisation of complete graphs, called a homogeneous factorisation. Characterising homogeneous factorisations is an important problem in graph theory with applications to Ramsey theory. This paper develops a framework for the study of TODs, establishes some numerical relations between the parameters involved in TODs, gives some reduction results with respect to the G-actions on Ω and on P, and gives some construction methods for TODs.

New Constructions of Self-Complementary Cayley Graphs

“…We remark, however, that a quotient graph Γ B of a self-complementary graph Γ is not necessarily self-complementary; refer to [16].…”

“…The first family of self-complementary vertex-transitive graphs that are not Cayley graphs was obtained by Li and Praeger [15] in 2001. After 2000, the study of self-complementary vertex-transitive graphs has been significantly advanced by the work in [11,16]. More recently, self-complementary vertex-transitive graphs of order pq where p, q are primes were classified [18] and self-complementary metacirculants were studied in [19].…”

Decompositions of the automorphism groups of edge-colored graphs into the direct product of permutation groups

“…For example, it leads to the concept of colored totally symmetric graphs, that was described in [11,12]. This coincides to a large extent with the research on homogeneous factorization of graphs (c.f., [4,15,16]). One direction of research is to consider various constructions of permutation groups and to ask the following question: is it true that if the components of the construction belong to a particular class G(k), then the result belongs to G(k), as well?…”

Edge‐transitive homogeneous factorizations of complete uniform hypergraphs

“…As generalizations of vertex‐transitive self‐complementary graphs, homogeneous factorizations of complete graphs (complete 2‐hypergraphs) were introduced in (and for graphs in general in ). The reader is referred to for the theory of homogeneous factorizations of graphs. In , Li, Lim, and Praeger classified the homogeneous factorizations of complete graphs with all factors admitting a common edge‐transitive group.…”