Balance, Partial Balance and Balanced-type Spectra in Graph-Designs

Abstract

For a given graph $G$, the set of positive integers $v$ for which a $G$-design exists is usually called the spectrum for $G$ and the determination of the spectrum is sometimes called the spectrum problem. We consider the spectrum problem for $G$-designs satisfying additional conditions of balance, in the case where $G$ is a member of one of the following infinite families of trees: caterpillars, stars, comets, lobsters, and trees of diameter at most $5$. We determine the existence spectrum for balanced $G$-designs, degree-balanced and partially degree-balanced $G$-designs, and orbit-balanced $G$-designs. We also address the existence question for non-balanced $G$-designs, for $G$-designs which are either balanced or partially degree-balanced but not degree-balanced, and for $G$-designs which are degree-balanced but not orbit-balanced.