On an extremal problem of regular graphs related to fractional repetition codes

Abstract

Fractional repetition (FR) codes are a special family of regenerating codes with the repair-by-transfer property. The constructions of FR codes are naturally related to combinatorial designs, graphs, and hypergraphs. Given the file size of an FR code, it is desirable to determine the minimum number of storage nodes needed. The problem is related to an extremal graph theory problem, which asks for the minimum number of vertices of an α-regular graph such that any subgraph with k vertices has at most δ edges. In this paper, we present a class of regular graphs for this problem to give the bounds for the minimum number of storage nodes for the FR codes.