Secret sharing on large girth graphs

We investigate graph based secret sharing schemes and its information ratio, also called complexity, measuring the maximal amount of information the vertices has to store. It was conjectured that in large girth graphs, where the interaction between far away nodes is restricted to a single path, this ratio is bounded. This conjecture was supported by several result, most notably by a result of Csirmaz and Ligeti (Computing 85(1):127–136, 2009) saying that the complexity of graphs with girth at least six and no neighboring high degree vertices is strictly below 2. In this paper we refute the above conjecture. First, a family of d-regular graphs is defined iteratively such that the complexity of these graphs is the largest possible (d + 1)/2 allowed by Stinson’s bound (IEEE Trans. Inf. Theory 40(1):118–125, 1994). This part extends earlier results of van Dijk (Des. Codes Crypt. 6(2):143–169, 1995) and Blundo et al. (Des. Codes Crypt. 11(2):107–110, 1997), and uses the so-called entropy method. Second, using combinatorial arguments, we show that this family contains graphs with arbitrary large girth. In particular, we obtain the following purely combinatorial result, which might be interesting on its own: there are d-regular graphs with arbitrary large girth such that any fractional edge-cover by stars (or by complete multipartite graphs) must cover some vertex (d + 1)/2 times.