Plane four-regular graphs with vertex-to-vertex unit triangles

AbstractFor the smallest number of non-overlapping vertex-to-vertex unit triangles in the plane it is proved ⩽42 in general, and ⩽3800 if additional triangles are not allowed

On h-perfect numbers

Let σ(x) denote the sum of the divisors of x. The diophantine equation σ(x) + σ(y) = 2(x + y) equalizes the abundance and deficiency of x and y. For x = n and y = hn the solutions n are called h-perfect since the classical perfect numbers occur as solutions for h = 1. Some results on h-perfect numbers are determined. Keywords: perfect numbers, amicable number

Independence on Triangular Triangle Boards

Triangular parts of the Euclidean triangle tessellation of the plane are considered as gameboards Tn. Thirteen chess-like pieces are defined as in [1]. The independence number βn is the maximum number of non-attacking copies of a piece on Tn. For nine of the chess-like pieces βn is determined completely