Leif Kjær Jørgensen

Contractions to K 8

It is proved that the maximal number of edges in a graph with n ≧ 8 vertices that is not contractible to K8 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 K8 are the K5(2)-cockades that have exactly 6n − 20 edges.

Non-existence of directed strongly regular graphs

Directed strongly regular graphs were introduced by Duval in 1988. We give several non-existence results, each excluding infinite series of feasible parameter sets: We prove a result that extends the absolute bound for strongly regular graphs, and we give a characterization of directed strongly regular graphs whose adjacency matrix has rank 3 or 4. We also give some combinatorial non-existence results and a construction of five new directed strongly regular graphs.

Diameters of cubic graphs

We prove that a graph with maximum degree 3 and diameter d≥4 cannot have exactly two vertices less than the Moore bound.

New mixed Moore graphs and directed strongly regular graphs

A 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 ) .

Girth 5 graphs from relative difference sets

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.

Directed strongly regular with m=l

Abstract Directed strongly regular graphs with μ = λ = t −1 and k −1 divisible by μ are constructed from cyclic groups. Nonexistence of a directed strongly regular graph with n=32, k=6, t=5, μ=λ=1 is proved.

On cages with given degree sets

Abstract We consider the problem of constructing minimal graphs of given girth having a particular degree set.

Vertex partitions of K4,4-minor free graphs

Abstract. We prove that a 4-connected K4,4-minor free graph on n vertices has at most 4n−8 edges and we use this result to show that every K4,4-minor free graph has vertex-arboricity at most 4. This improves the case (n,m)=(7,3) of the following conjecture of Woodall: the vertex set of a graph without a Kn-minor and without a -minor can be partitioned in n−m+1 subgraphs without a Km-minor and without a -minor.

Nonexistence of certain cubic graphs with small diameters

Abstract We consider the maximum number of vertices in a cubic graph with small diameter. We show that a cubic graph of diameter 4 has at most 40 vertices. (The Moore bound is 46 and graphs with 38 vertices are known.) We also consider bipartite cubic graphs of diameter 5, for which the Moore bound is 62. We prove that in this case a graph with 56 vertices found by Bond and Delorme (1988) is optimal.

Algorithmic Approach to Non-symmetric 3-class Association Schemes

There are 24 feasible parameter sets for a primitive non-symmetric association schemes with 3 classes and at most 100 vertices. Using computer search, we prove non-existence for three feasible parameter sets. Ten cases are still open.