A (v, b, r, k) combinatorial configuration can be defined as a connected, (r, k)-biregular bipartite graph with v vertices on one side and b vertices on the other 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 semigroup is completely described whenever r = 2 or r = 3.
For the remaining cases some bounds are given on the multiplicity and the conductor of the numerical semigroup. This leads to
some concluding results on the existence of configurable tuples.
CitationBras Amorós, Maria; Stokes, Klara. On the existence of combinatorial configurations. A: International Workshop on Optimal Networks Topologies. "Proceedings of the 3rd International Workshop on Optimal Networks Topologies IWONT 2010". Barcelona: Iniciativa Digital Politècnica, 2011, p. 145-167.
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