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

Abstract

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 $4k+2$.

Figures (3)

• Figure 1: $\mathcal{D}_k$ (The solid arcs represent the arcs and the dashed arcs represent the paths.)
• Figure 2: The structure of $C$ in $D$. The solid arcs represent the arcs of $D$ and the dashed arcs represent the paths of $D$.
• Figure 3: The structure of $C$ in $D$. The solid arcs represent the arcs of $D$ and the dashed arcs represent the directed paths of $D$. We demonstrate the situation $X_{h-1}\cap X_h\neq \emptyset$ in this figure.

Theorems & Definitions (10)

• Conjecture 1.1
• proof
• proof : Proof of Claim \ref{['cut']}.
• proof : Proof of Claim \ref{['order']}.
• proof : Proof of Claim \ref{['no cut']}.
• proof : Proof of Claim \ref{['dis']}.
• proof : Proof of Claim \ref{['wlog']}.
• proof : Proof of Claim \ref{['intersect_1']}.
• proof : Proof of Claim \ref{['intersect_2']}.
• proof : Proof of Claim \ref{['main']}.