# On the spanning connectivity of tournaments

## TL;DR

This paper investigates the spanning connectivity properties of tournaments, establishing lower bounds for strong and weak spanning connectivity based on the number of vertices and irregularity, thus advancing understanding of digraph connectivity.

## Contribution

It provides new bounds on the strong and weak spanning connectivity of tournaments with given irregularity and size, extending previous connectivity results in digraph theory.

## Key findings

- For tournaments with irregularity at most k, spanning connectivity bounds depend on the number of vertices.
- Established that $
abla_{s}^{*}(T) ext{ and }
abla_{w}^{*}(T)$ have specific lower bounds based on size and irregularity.
- Results apply to large tournaments with specified irregularity constraints.

## Abstract

Let $D$ be a digraph. A $k$-container of $D$ between $u$ and $v$, $C(u,v)$, is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u,v)$ of $D$ is a strong (resp. weak) $k^{*}$-container if there is a set of $k$ internally disjoint paths with the same direction (resp. with different directions allowed) between $u$ and $v$ and it contains all vertices of $D$. A digraph $D$ is $k^{*}$-strongly (resp. $k^{*}$-weakly) connected if there exists a strong (resp. weak) $k^{*}$-container between any two distinct vertices. We define the strong (resp. weak) spanning connectivity of a digraph $D$, $\kappa_{s}^{*}(D)$ (resp. $\kappa_{w}^{*}(D)$ ), to be the largest integer $k$ such that $D$ is $\omega^{*}$-strongly (resp. $\omega^{*}$-weakly) connected for all $1\leq \omega\leq k$ if $D$ is a $1^{*}$-strongly (resp. $1^{*}$-weakly) connected. In this paper, we show that a tournament with $n$ vertices and irregularity $i(T)\leq k$, if $n\geq6t+5k$ $(t\geq2)$, then $\kappa_{s}^{*}(T)\geq t$ and $\kappa_{w}^{*}(T)\geq t+1$ if $n\geq6t+5k-3$ $(t\geq2)$.