Weak sequenceability in cyclic groups

A subset (Formula presented.) of an abelian group (Formula presented.) is sequenceable if there is an ordering (Formula presented.) of its elements such that the partial sums (Formula presented.), given by (Formula presented.) and (Formula presented.) for (Formula presented.), are distinct, with the possible exception that we may have (Formula presented.). In the literature there are several conjectures and questions concerning the sequenceability of subsets of abelian groups, which have been combined and summarized by Alspach and Liversidge into the conjecture that if a subset of an abelian group does not contain 0 then it is sequenceable. If the elements of a sequenceable set (Formula presented.) do not sum to 0 then there exists a simple path (Formula presented.) in the Cayley graph (Formula presented.) such that (Formula presented.). In this paper, inspired by this graph–theoretical interpretation, we propose a weakening of this conjecture. Here, under the above assumptions, we want to find an ordering whose partial sums define a walk (Formula presented.) of girth bigger than (Formula presented.) (for a given (Formula presented.)) and such that (Formula presented.). This is possible given that the partial sums (Formula presented.) and (Formula presented.) are different whenever (Formula presented.) and (Formula presented.) are distinct and (Formula presented.). In this case, we say that the set (Formula presented.) is (Formula presented.) -weakly sequenceable. The main result here presented is that any subset (Formula presented.) of (Formula presented.) is (Formula presented.) -weakly sequenceable whenever (Formula presented.) or when (Formula presented.) does not contain pairs of type (Formula presented.) and (Formula presented.).