Postman problems on series–parallel mixed graphs

Abstract The undirected degree/diameter and degree/girth problems and their directed analogues have been studied for many decades in the search for efficient network topologies. Recently such questions have received much attention in the setting of mixed graphs, i.e. networks that admit both undirected edges and directed arcs. The degree/diameter problem for mixed graphs asks for the largest possible order of a network with diameter k, maximum undirected degree $$\le r$$≤r and maximum directed out-degree $$\le z$$≤z. Similarly one can search for the smallest possible k-geodetic mixed graphs with minimum undirected degree $$\ge r$$≥r and minimum directed out-degree $$\ge z$$≥z. A simple counting argument reveals the existence of a natural bound, the Moore bound, on the order of such graphs; a graph that meets this limit is a mixed Moore graph. Mixed Moore graphs can exist only for $$k = 2$$k=2 and even in this case it is known that they are extremely rare. It is therefore of interest to search for graphs with order one away from the Moore bound. Such graphs must be out-regular; a much more difficult question is whether they must be totally regular. For $$k = 2$$k=2, we answer this question in the affirmative, thereby resolving an open problem stated in a recent paper of López and Miret. We also present partial results for larger k. We finally put these results to practical use by proving the uniqueness of a 2-geodetic mixed graph with order exceeding the Moore bound by one.

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Mixed Moore Cayley Graphs

Journal of Interconnection Networks ◽

10.1142/s0219265917410109 ◽

2017 ◽

Vol 17 (03n04) ◽

pp. 1741010

Author(s):

GRAHAME ERSKINE

Keyword(s):

Infinite Number ◽

Upper Bound ◽

Maximum Degree ◽

Cayley Graphs ◽

Computational Techniques ◽

Elementary Group ◽

Recent Interest ◽

Mixed Graphs ◽

Rule Out

The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected edges and directed arcs in the graph. For a diameter 2 graph with maximum undirected degree r and directed out-degree z, a straightforward counting argument yields an upper bound M(z, r, 2) = (z+r)2+z+1 for the order of the graph. Apart from the case r = 1, the only three known examples of mixed graphs attaining this bound are Cayley graphs, and there are an infinite number of feasible pairs (r, z) where the existence of mixed Moore graphs with these parameters is unknown. We use a combination of elementary group-theoretical arguments and computational techniques to rule out the existence of further examples of mixed Cayley graphs attaining the Moore bound for all orders up to 485.

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