On the Existence of $(K_5 \setminuse)$-Designs with Application to Optical Networks

Abstract

Motivated by the connection between graph decompositions and traffic grooming in optical networks, we continue the investigation of the existence problem for $(K_5 \setminus e)$-designs of order n. It is proved that the necessary conditions for the existence of such designs are also sufficient with 3 definite exceptions $(n=9,10,18)$ and 12 possible exceptions with $n=234$ being the largest. This gives a near solution for the long standing problem posed by Bermond et al. in [Ars Combin., 10 (1980), pp. 211–254]. As a consequence, we also give an optimal grooming on n nodes with $C=9$ when such a $(K_5 \setminus e)$-design of order n exists.