The Semigroup of Combinatorial Configurations

TL;DR

This paper studies the set of combinatorial configurations represented by bipartite graphs with specific properties, showing that the set of feasible parameters forms a numerical semigroup, fully characterized for certain cases.

Contribution

It proves that the set of configurable tuples with fixed r and k forms a numerical semigroup and provides a complete description for r=2 and r=3.

Findings

The set of configurable tuples forms a numerical semigroup.

Complete characterization of the semigroup for r=2.

Complete characterization of the semigroup for r=3.

Abstract

A (v,b,r,k) combinatorial configuration is a (r,k)-biregular bipartite graph with v vertices on the left and b vertices on the right and with no cycle of length 4. Combinatorial configurations have become very important for some cryptographic applications to sensor networks and to peer-to-peer communities. Configurable tuples are those tuples (v,b,r,k) for which a (v,b,r,k) combinatorial configuration exists. It is proved in this work that the set of configurable tuples with fixed r and k has the structure of a numerical semigroup. The numerical semigroup is completely described for r=2 and r=3.