On Networks with Order Close to the Moore Bound

Abstract

The degree/diameter problem for mixed graphs asks for the largest possible order of a mixed graph with given diameter and degree parameters. Similarly the degree/geodecity problem concerns the smallest order of a k-geodetic mixed graph with given minimum undirected and directed degrees; this is a generalisation of the classical degree/girth problem. In this paper we present new bounds on the order of mixed graphs with given diameter or geodetic girth and exhibit new examples of directed and mixed geodetic cages. In particular, we show that any k-geodetic mixed graph with excess one must have geodetic girth two and be totally regular, thereby proving an earlier conjecture of the authors.

The structure of digraphs with excess one

“…Further results on the problem and examples of geodetic cages can be found in [23–26]. As no nontrivial digraphs with excess one have been found, we make the following conjecture.…”

“…These results may suggest that the permutation digraphs are smallest possible arc‐transitive $$k$$‐geodetic digraphs for sufficiently large $$d$$ [26].…”

“…It was shown in [26] that for any $$d,k\ge 1$$ there exists a diregular $$k$$‐geodetic digraph with degree $$d$$, so this problem is well‐posed.…”

Moore mixed graphs from Cayley graphs

“…For the case r = z = 1, Tuite and Erskine [21] gave an improved bound for Theorem 2.3. For optimal (1, 1)-regular mixed graphs with diameter 3, we have the following result.…”