A note on randomly complete -partite graphs
For any positive integer k and any set A of nonnegative integers, let denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both and hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying for all n ≥ n₀, we have as n → ∞.
We prove that a rank Dowling geometry of a group is partition representable if and only if is a Frobenius complement. This implies that Dowling group geometries are secret-sharing if and only if they are multilinearly representable.
A graph , with a group of automorphisms of , is said to be -transitive, for some , if is transitive on -arcs but not on -arcs. Let be a connected -transitive graph of prime valency , and the vertex stabilizer of a vertex . Suppose that is solvable. Weiss (1974) proved that . In this paper, we prove that for some positive integers and such that and .
Strongly perfect graphs were introduced by C. Berge and P. Duchet in [1]. In [4], [3] the following was studied: the problem of strong perfectness for the Cartesian product, the tensor product, the symmetrical difference of n, n ≥ 2, graphs and for the generalized Cartesian product of graphs. Co-strong perfectness was first studied by G. Ravindra andD. Basavayya [5]. In this paper we discuss strong perfectness and co-strong perfectness for the generalized composition (the lexicographic product)...
In this note we give an upper bound for λ(n), the maximum number of edges in a strongly multiplicative graph of order n, which is sharper than the upper bound obtained by Beineke and Hegde [1].