An improvement bound on a problem of Picasarri-Arrieta and Rambaud

Let $k$ and $\ell$ be positive integers. A cycle with two blocks $C(k,\ell)$ is a digraph consisting of two internally vertex disjoint directed paths of lengths $k$ and $\ell$ with the same initial vertex and terminal vertex. Picasarri-Arrieta and Rambaud (European J. Combin., 2024) proved that for any $k\geq 2$, every digraph of minimum out-degree at least two and girth at least $8k-6$ contains a subdivision of $C(k,k)$. They also construct a family of digraphs showing that the girth cannot be reduced to $k-1$, and posed the problem of determining the minimum girth such that every digraph of minimum out-degree at least two contains a subdivision of $C(k,k)$. In this paper, we improve the lower bound on the girth from $8k-6$ to $4k+2$, and construct a family of digraphs in which every member has minimum out-degree two and girth $k$ but contains no subdivision of $C(k,k)$. Thus our results show that the girth in question lies between $k+1$ and $4k+2$.