Some Problems On Zero-sum Sequences

Zero-sum theory is a subfield of combinatorial number theory, and there are intrinsic connections with graph theory, Ramsey theory, geometry and number theory etc.. The main object that we study in zero-sum theory is zero-sum sequence, which is a sequence in an additive finite abelian group with the sum zero.In this paper, according to two different points of view of direct zero-sum problem and inverse zero-sum problem, the author gets some results on several combinatorial constants and the stucture of some extremal sequences.Two direct zero-sum problems are considered in Chapter Two and Chapter Three, which includes several combinatorial constants r(G),D(G),s(G) and η(G). In Chapter Two, the author extends the relationship r(G) = |G| + D(G) - 1 in finite abelian groups to some type of finite non-abelian groups. In Chapter Three, for a special type of abelian p-groups G, the author gets the exact value of s(G) and η(G) and verifies the conjecture s(G) = η(G) + exp(G) - 1.In Chapter Four, the author defines normal sequences and unextendible sequences, and studies the structure of these two types of sequences from the point of view of inverse zero-sum problems. The structure of the normal sequences in finite cyclic groups, some elementary p-groups and groups of the form G = C_n⊕C_n where n has property B was determined exactly. In addition, the author gets close relations beteen the structure of the extremal sequence S in an abelian group G, where S satisfies |S| = |G|+ D(G) - 2 and 0 ? Σ_|G|(S), and tne structure of normal sequences. Similarly, the author obtains some characters for unextendible sequences in finite cyclic groups, the groups with rank 2 and general abelian groups.In Chapter Five, the author considers two zero-sum problems in finite cyclic groups, that is the equivalence classes of minimal zero-sum sequences and the weighted sequences. An upper bound for a type of equivalence class sequences was obtained, and for a conjecture about weighted sequences arosed by A. Bialostocki in CANT2005, the author provides a proof when the group is a cyclic group with prime order and the multiplicities of each element in the two sequences are not very large.