On complete multipartite derangement graphs

Abstract

Given a finite transitive permutation group G ≤ Sym(Ω), with |Ω| ≥ 2, the derangement graph Γ G of G is the Cayley graph Cay(G, Der(G)), where Der(G) is the set of all derangements of G. Meagher et al. [On triangles in derangement graphs, J. Combin. Theory Ser. A, 180:105390, 2021] recently proved that Sym(2) acting on {1, 2} is the only transitive group whose derangement graph is bipartite and any transitive group of degree at least three has a triangle in its derangement graph. They also showed that there exist transitive groups whose derangement graphs are complete multipartite.This paper gives two new families of transitive groups with complete multipartite derangement graphs. In addition, we prove that if p is an odd prime and G is a transitive group of degree 2p, then the independence number of Γ G is at most twice the size of a point-stabilizer of G.

On intersection density of transitive groups of degree a product of two odd primes

“…Conjectures 1.1(ii) and (iv) were settled, respectively, in [6] and [17]. Furthermore in [6] it was shown that I 2p = {1, 2} for every odd prime p and a complete characterization of groups of degree 2p with intersection density 2 was also given there.…”

On the intersection density of the symmetric group acting on uniform subsets of small size

“…For arbitrary transitive groups, however, intersecting sets can have size larger than the order of a stabilizer of a point and the maximum intersecting sets can have a more complex structure than the canonical intersecting sets (see [25] for instance). The smallest example of transitive groups (both in terms of order and degree) whose maximum intersecting sets have size larger than the order of stabilizer of a point is the alternating group Alt(4) acting on the 2-subsets of {1, 2, 3, 4} (see [29]).…”

“…Moreover, if ρ(G) = k > 1, then the largest intersecting set of G has size k times the order of a point stabilizer of G (i.e., the size of the canonical intersecting sets). Several papers on the intersection density of transitive groups have recently appeared in the literature [17,18,19,22,29,30].…”

Intersection density of transitive groups of certain degrees

“…Following [12] we define I n to be the set of all intersection densities of transitive permutation groups of degree n, that is, Conjecture 1.2 is settled in [13], where an additional problem regarding the possible values of intersection densities in I 2p was posed.…”