A variant of the McKay-Miller-Širáň construction for mixed graphs

Abstract

The Degree/Diameter Problem is an extremal problem in graph theory with applications in network design. One of the main research areas in the Degree/Diameter Problem consists of finding large graphs whose order approach the theoretical upper bounds as much as possible. In the case of directed graphs there exist some families that come close to the theoretical upper bound asymptotically. In the case of undirected graphs there also exist some good constructions for specific values of the parameters involved (degree and/or diameter). One such construction was given by McKay, Miller, and Širáň in [McKay, B., M. Miller and J. Širáň, A note on large graphs of diameter two and given maximum degree, J Comb Theo Ser B 74 (1998), 110-118], which produces large graphs of diameter 2 with the aid of the voltage assignment technique. Here we show how to re-engineer the McKay-Miller-Širáň construction in order to obtain large mixed graphs of diameter 2, i.e. graphs containing both directed arcs and undirected edges.