On Blocking Sets in Almost Balanced Path Designs

Let \(r(a)\) be the replication number of the vertex \(a\) of a path design \(P(v,k, 1)\), \(k \geq 3\). Let \(\bar{r}(v,k) = \text{min}\{\text{max}_{a\in V} \,r(a) | (V,\mathcal{B}) \text{ is a } P(v,k, 1)\}\). A path design \(P(v,k,1)\), \((W,\mathcal{D})\), is said to be \emph{almost balanced} if \(\bar{r}(v,k) – 1 \leq r(y) \leq \bar{r}(v,k)\) for each \(y \in W\). Let \(v \equiv 0 \text{ or } 1 \pmod{2(k-1)}\) (for each odd \(k\), \(k \geq 3\)) and let \(v_y \equiv 0 \text{ or } 1 \pmod{k-1}\) (for each even \(k\), \(k \geq 4\)). 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 \(x\).