2018/07/11 by Niranjan Balachandran, Balachandran, Niranjan, Eshita Mazumdar +1
Mathematics · #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1807.04112
For a finite abelian group G written additively, and a non-empty subset\nA\⊂ [1,\exp(G)-1] the weighted Davenport Constant of G with respect to\nthe set A, denoted DA(G), is the least positive integer k for which the\nfollowing holds: Given an arbitrary G-sequence (x1,\…,xk), there\nexists a non-empty subsequence (xi1,\…,xit) along with aj\∈\nA such that \∑j=1t ajxij=0. In this paper, we pose and study a\nnatural new extremal problem that arises from the study of DA(G): For an\ninteger k\≥ 2, determine fDG(k):=\min |A|: DA(G)\≤ k (if the\nproblem posed makes sense). It turns out that for k `not-too-small', this is\na well-posed problem and one of the most interesting cases occurs for G= Zp,\nthe cyclic group of prime order, for which we obtain near optimal bounds for\nall k (for sufficiently large primes p), and asymptotically tight (up to\nconstants) bounds for k=2,4.\n