
Let
are sufficient to guarantee the existence of such a Latin rectangle.
For positive integers
A necessary and sufficient condition for a family of finite sets to possess a collection of
Let
The foundation of an analytic graph theory is developed.
The integrity of a graph,
It is shown that unlike the chromatic polynomial, which does not characterize unions of non-trivial graphs, the circuit polynomial characterizes the unions of many families of graphs. They include unions of chains, cycles and mixtures of these graphs, also unions of complete graphs. It is also shown that in general, if a Hamiltonian graph is characterized by its circuit polynomial, then so also is the union of the graph with itself.
In this paper, we obtain results on the number of constraints
A basis is exhibited for the first homology space of a surface over a field. This basis is found by extending a basis of the boundary cycle space of an embedded graph to the cycle space of the graph.
Let
An undirected graph of diameter
Lander Conjectured: If
In a previous paper, the above conjecture was verified for
Generalized Moore graphs are regular graphs that satisfy an additional distance condition, namely, that there be the maximum number of vertices as close as possible to any particular vertex, when that vertex is considered as root vertex. These graphs form a useful model for the study of various theoretical properties of computer communications networks. In particular, they lend themselves to a discussion of lower bounds for network cost, delay, reliability, and vulnerability. A considerable number of papers have already been published concerning the existence and properties of generalized Moore graphs of valence three, and some initial studies have discussed generalized Moore graphs of valence four, when the number of vertices is less than fourteen. This paper continues the previous studies for those cases when the graph contains a number of vertices that is between fourteen and twenty. In the case of valence three, the graph with a complete second level exists; it is just the Petersen graph. The situation is quite different for valence four; not only does the graph with a complete second level not exist, but the graphs in its immediate “neighbourhood” also fail to exist.
In this paper, we investigate the existence of skew frames with sets of skew transversals. We consider skew frames of type
All graphs meeting the basic necessary conditions to be the leave graph of a maximal partial triple system with at most thirteen elements are generated. A hill-climbing algorithm is developed to determine which of these candidates are in fact leave graphs. Improved necessary conditions for a graph to be a leave graph are developed.
Some new lower bounds for higher Ramsey numbers are presented. Results concerning generalized hypergraph Ramsey numbers are also given.
1970-2025 CP (Manitoba, Canada) unless otherwise stated.