On Blocking Sets in Almost Balanced Path Designs

Abstract

Let $r(a)$ be the replication number of the vertex $a$ of a path design $P(v,k,1)$, $k\ge 3$. Let $\overline{r}(v,k)=\text{min}\{{\text{max}}_{a\in V}r(a)|(V,\mathcal{B})\text{ is a }P(v,k,1)\}$. A path design $P(v,k,1)$, $(W,\mathcal{D})$, is said to be $almostbalanced$ if $\overline{r}(v,k)–1\le r(y)\le \overline{r}(v,k)$ for each $y\in W$. Let $v\equiv 0\text{ or }1(\mathrm{mod}2(k-1))$ (for each odd $k$, $k\ge 3$) and let $v_y\equiv 0\text{ or }1(\mathrm{mod}k-1)$ (for each even $k$, $k\ge 4$). In this note, we determine the spectrum $\mathcal{B}\mathcal{S}\mathcal{A}\mathcal{B}\mathcal{P}(v,k,1)$ of integers $x$ such that there exists an almost balanced path design $P(v,k,1)$ with a blocking set of cardinality $x$.