It is proved that the maximal number of edges in a graph with n L 8 vertices that is not contractible to & is 6n -21, unless 5 divides n, and the only graphs with n = 5m vertices and more than 6n -21 edges that are not contractible to KS are the K5(2)-cockades that have exactly 6n -20 edges. 0 1994 John Wiley & Sons, Inc.is a Ks(2)-cockade.If a graph G is contractible to a complete graph K,,, then one of the simplicia1 summands is contractible to K,,. Therefore Ks(2)-cockades are not By the above definition, a K5(2)-cockade, (MPi-cockade) is the union of some spanned subgraphs that are isomorphic to K5(2) (graphs obtained by adding i vertices to a K4 or a 4-connected maximal planar graph). These unique subgraphs are called the cockade-members.A cockade-member of a Kg(2)-cockade (MPi-cockade) is an end-member if it has at most one K5 (K3+i) that is also contained in another cockademember. CONTRACTIONS TO Ka 433The minimum degree and the maximum degree of a graph G is denoted Otherwise we use standard notation and terminology, see Bollob6s [3].

a b s t r a c tA directed strongly regular graph with parameters (n, k, t, λ, µ) is a k-regular directed graph with n vertices satisfying that the number of walks of length 2 from a vertex x to a vertex y is t if x = y, λ if there is an edge directed from x to y and µ otherwise. If λ = 0 and µ = 1 then we say that it is a mixed Moore graph. It is known that there are unique mixed Moore graphs with parameters (k 2 + k, k, 1, 0, 1), k ≥ 2, and (18, 4, 3, 0, 1). We construct a new mixed Moore graph with parameters (108, 10, 3, 0, 1) and also new directed strongly regular graphs with parameters (36, 10, 5, 2, 3) and (96, 13, 5, 0, 2). This new graph on 108 vertices can also be seen as an example of a so called multipartite Moore digraph. Finally we consider the possibility that mixed Moore graphs with other parameters could exist, in particular the first open case which is (40, 6, 3, 0, 1).

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We consider the problem of construction of graphs with given degree k and girth 5 and as few vertices as possible. We give a construction of a family of girth 5 graphs based on relative difference sets. This family contains the smallest known graph of degree 8 and girth 5 which was constructed by Royle, four of the known cages including the Hoffman-Singleton graph, some graphs constructed by Exoo and some new smallest known graphs.

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