For n > r > 1, let f,(n) denote the minimum number q, such that it is possible to partition all edges of the complete r-graph on n vertices into q complete r-partite r-graphs. Graham and Pollak showed that fz(n) = n-1. Here we observe that f3(n) = n-2 and show that for every fixed r > 2, there are positive constants cx(r) and c2(r) such that q(r) < f,(n)" n-f'/2J < c2(r) for all n > r. This sol...

For finite graphs F and G, let Nr(G) denote the number of occurrences of F in G, i.e., the number of subgraphs of G which are isomorphic to F. I f g and c?? are families of graphs, it is natural to ask then whether or not the quantities NF(G), FE F, are linearly independent when G is restricted to Q. For example, if P = (K1, &} (where K, denotes the complete graph on n vertices) and 9 is the fa...

In this paper, in our modification of Graham scan for determining the convex hull of a finite planar set, we show a restricted area of the examination of points and its advantage. The actual run times of our scan and Graham scan on the set of random points shows that our modified algorithm runs significantly faster than Graham’s one.

A particular case of a conjecture of Erdös and Graham, which concerns the number of integer points on a family of quartic curves, is investigated. An absolute bound for the number of such integer points is obtained.

In a well-known result, Ronald Graham found a Fibonacci-like sequence whose two initial terms are relatively prime and which consists only of composite integers. We generalize this result to nondegenerate second-order recurrences.

Let P be a point set with n elements in general position. A triangulation T of P is a set of triangles with disjoint interiors such that their union is the convex hull of P , no triangle contains an element of P in in its interior, and the vertices of the triangles of T are points of P . Given T we define a graph G(T ) whose vertices are the triangles of T , two of which are adjacent if they sh...

A T -decomposition of a graph G is a set of edge-disjoint copies of T in G that cover the edge set of G. Graham and Häggkvist (1989) conjectured that any 2l-regular graph G admits a T -decomposition if T is a tree with l edges. Kouider and Lonc (1999) conjectured that, in the special case where T is the path with l edges, G admits a T -decomposition D where every vertex of G is the end-vertex o...

The study of the representation theory of the symmetric groups (and, more recently, the Iwahori–Hecke algebras of type A) has always been inextricably linked with the combinatorics of partitions. More recently, the complex reflection group of type G(r, 1, n) and its Hecke algebras (the Ariki–Koike algebras or cyclotomic Hecke algebras) have been studied, and it is clear that there is a similar ...

We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of [1, n], of monochromatic {x, y, x + ay} triples for a ≥ 1. We give a new proof of the original case of a = 1. We show that the minimum number of such triples is at most n 2 2a(a2+2a+3) + O(n) when a ≥ 2. We also find a new upper bound for the minimum number, over all r-colorings of [1, n], of monochrom...

A cycle C = {v1, v2, . . . , v1} in a tournament T is said to be even, if when walking along C, an even number of edges point in the wrong direction, that is, they are directed from vi+1 to vi. In this short paper, we show that for every fixed even integer k ≥ 4, if close to half of the k-cycles in a tournament T are even, then T must be quasi-random. This resolves an open question raised in 19...