Jens-P. Bode

Hexagonal polyomino achievement

Abstract The cells of a tessellation as game board are alternately marked by players A and B. A given polyomino P is a winner if A can achieve P in his marks regardless of the moves of B. Otherwise it is called a loser. For the hexagonal tessellation all but five polyominoes with at most five cells are determined as winners or losers. There are at most 13 938 winners altogether and no winner has more than 18 cells.

The minimum size of k-rainbow connected graphs of given order

Abstract Let t ( n , k ) denote the minimum number of edges of a k -rainbow connected graph G of order n , that is a graph G for which a coloring of the edges with at most k colors exists such that any two vertices of G are connected by a path with edges of pairwise different colors. A general upper bound of t ( n , k ) and the exact value of t ( n , 3 ) are given.

Directed paths of diagonals within polygons

It is proved that within an n-gon there exists a directed path using diagonals of all n-1 lengths exactly once excluded an arbitrarily fixed length.

Independence for knights on hexagon and triangle boards

For three kinds of chess-like knights, one on triangle and two on hexagon boards, the independence numbers for their knight graphs are considered. We determine these numbers completely for two kinds of these knights, and for one residue class modulo 4 in the case of the third kind.

Some Ramsey Schur Numbers

The Ramsey Schur number

Positive Integers (a2 + b2)/(ab + 1) are Squares

RS(s,t)

Improved Sufficient Conditions for Hamiltonian Properties

is the smallest

Achievement Games for Polyominoes on Archimedean Tessellations

n

King independence on triangle boards

such that every 2-colouring of the edges of

Hypercube-Anti-Ramsey Numbers of Q 5 .

K_n