Given a graph Γ, a subset C of V (Γ) is called a perfect code in Γ if every vertex of Γ is at distance no more than one to exactly one vertex in C, and a subset C of V (Γ) is called a total perfect code in Γ if every vertex of Γ is adjacent to exactly one vertex in C. In this paper we study perfect codes and total perfect codes in Cayley graphs, with a focus on the following themes: when a subgroup of a given group is a (total) perfect code in a Cayley graph of the group; and how to construct new (total) perfect codes in a Cayley graph from known ones using automorphisms of the underlying group. We prove several results around these questions.

a b s t r a c tIn this paper, we present improved bounds for the First Fit algorithm for the bin-packing problem. We prove C FF (L) ≤ 17 10 C * (L)+ 7 10 for all lists L, and the absolute performance ratio of FF is at most 12 7 .

The automorphism group of a flag-transitive 6-(v, k, 2) design is a 3-homogeneous permutation group. Therefore, using the classification theorem of 3-homogeneous permutation groups, the classification of flag-transitive 6-(v, k,2) designs can be discussed. In this paper, by analyzing the combination quantity relation of 6-(v, k, 2) design and the characteristics of 3-homogeneous permutation groups, it is proved that: there are no 6-(v, k, 2) designs D admitting a flag transitive group G ≤ Aut (D) of automorphisms.

A characterization is given for the factorizations of almost simple groups with a solvable factor. It turns out that there are only several infinite families of these nontrivial factorizations, and an almost simple group with such a factorization cannot have socle exceptional Lie type or orthogonal of minus type. The characterization is then applied to study s s -arc-transitive Cayley graphs of solvable groups, leading to a striking corollary that, except for cycles, a non-bipartite connected 3 3 -arc-transitive Cayley graph of a finite solvable group is necessarily a normal cover of the Petersen graph or the Hoffman-Singleton graph.

We study vertex-quasiprimitive 2-arc-transitive digraphs, reducing the problem of vertex-primitive 2-arc-transitive digraphs to almost simple groups. This includes a complete classification of vertex-quasiprimitive 2-arc-transitive digraphs where the action on vertices has O'Nan-Scott type SD or CD.