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In this paper we introduce a new way of measuring the robustness Erd\H{o}s--Ko--Rado (EKR) Theorems on permutation groups. EKR-type results can be viewed as about independence numbers certain corresponding graphs, namely derangement and random subgraphs these graphs have been used to measure extremal results. context groups, are Cayley group in question. We propose studying properties that themselves group, robustness. present variety EKR property various groups using measure.

We generalize a classical result of Sabidussi that was improved by Hemminger, to the case directed color graphs. The original results give necessary and sufficient condition on two graphs, $C$ $D$, for automorphsim group wreath product ${\rm Aut}(C\wr D)$ be automorphism groups Aut}(C)\wr {\rm Aut}(D)$. Their characterization generalizes directly but we show there are additional exceptional cases in which either or $D$ is an infinite graph. Also, determine what Aut}(C \wr if D) \neq Aut} (C)...

We prove that if Cay( G ; S ) is a connected Cayley graph with n vertices, and the prime factorization of very small, then has hamiltonian cycle. More precisely, p , q r are distinct primes, can be form kp 24 ≠ k < 32, or kpq ≤ 5, pqr 2 4, 3 2.

We attempt to determine the structure of automorphism group a generic circulant graph. first show that almost all graphs have groups as small possible. The second author has conjectured remaining (di)graphs (those whose is not possible) are normal (di)graphs. this conjecture true in general, but if we consider only those order "large" subset integers. note non-normal can be classified into two natural classes (generalized wreath products, and deleted type), neither these contains every digraph.

This is the third, and last, of a series papers dealing with oriented regular representations. Here we complete classification finite groups that admit an representation (or ORR for short), give answer to 1980 question László Babai: 'Which [finite] graph as DRR'? It easy see well understood generalised dihedral do not s. We prove that, 11 small exceptions (having orders ranging from 8 64), every group has .

We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups G, such every automorphism connected on G has very simple form: composition left-translation and group automorphism. find classes have property, determine orders all do not property. also analogous results for permute colours, rather than preserving them.

In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs, using normal quotient reduction. We show that irreducible graphs in family have quasiprimitive automorphism groups, prove (using Classification Finite Simple Groups) no graph has a holomorphic simple group. also find some constraints on parameters reduce to complete graphs.

Suppose G is a finite group, such that |G| = 16 p , where prime. We show if S any generating set of then there hamiltonian cycle in the corresponding Cayley graph Cay( ; ).