Tree-designs with balanced-type conditions

For a given graph G we say that a G -design is balanced if there exists a constant r such that for each point x the number of blocks containing x is equal to r . esign is degree-balanced if, for each degree d occurring in the graph G , there exists a constant r d such that, for each point x , the number of blocks containing x as a vertex of degree d is equal to r d . 1 , V 2 , …, V h be the vertex-orbits of G under its automorphism group. A G -design is said to be orbit-balanced (or strongly balanced) if for i = 1 , 2 ,…, h there exists a constant R i such that, for each point x the number of blocks of the G -design in which x occurs as an element in the orbit V i is equal to R i . s a tree with six vertices, we determine the values of v for which a balanced G -design with v points exists, the values of v for which a degree-balanced G -design with v points exists, and the values of v for which an orbit-balanced G -design with v points exists. We also consider the existence problem for G -designs which are not balanced, which are balanced but not degree-balanced, and which are degree-balanced but not orbit-balanced.