Ingrid Mengersen

Matching-star Ramsey sets

Abstract The Ramsey set R (G,H) consists of all graphs F with F→(G,H) and F′↛(G,H) for every proper subgraph F′ of F. In this paper we will characterize the graphs belonging to R (2K 2 ,K 1,n ) with n⩾3 and determine R (2K 2 ,K 1,n ) for n⩽3 explicitly.

Edges without crossings in drawings of complete graphs

In drawings (two edges have at most one point in common, either a node or a crossing) of the complete graph Kn in the Euclidean plane there occur at most 2n − 2 edges without crossings. This was proved by G. Ringel in [1]. Here the minimal number of edges without crossings in drawings of Kn is determined, and for the existence of values between minimum and maximum is asked.

All Ramsay numbers for 5 vertices and 7 or 8 edges

Abstract For five vertices there are four graphs with seven edges and two graphs with eight edges. For all these six graphs the exact Ramsey numbers are given. Hence, for graphs with at most five vertices only the Ramsey number of the complete graph K 5 remains unknown.

Bounds on Ramsey Numbers of Certain Complete Bipartite Graphs

Bounds on the Ramsey number r(Kl,m,Kl,n), where we may assume l ≤ m ≤ n, are determined for 3 ≤ l ≤ 5 and m ≈ n. Particularly, for m = n the general upper bound on r(Kl,n, Kl,n) due to Chung and Graham is improved for those l. Moreover, the behavior of r(K3,m, K3,n) is studied for m fixed and n sufficiently large.

Ramsey numbers in octahedron graphs

Abstract The octahedron Ramsey number r O = r O ( G 1 ,…, G t ) is introduced as the smallest n such that any t-coloring of the edges of the octahedron graph O n = K 2 n − nK 2 contains for some i a subgraph G i of color i. With r = r ( G 1 ,…, G t ) denoting the classical Ramsey number, r O is between r /2 and r. If all G i s are complete, then r O = r . If all G i s are certain stars, then r O =⌈ r /2⌉. For all G i with at most four vertices, all values r O ( G 1 , G 2 ) are listed.

Point sets with many unit circles

Abstract Every three of n points in the plane determine a circle. The maximum number f ( n ) of congruent circles is determined for n ⩽ 7: f (3) = 1, f (4) = f (5) = 4, f (6) = 8, f (7) = 12.