On the spanning connectivity of tournaments

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 $n\geq6t+5k-3$ $(t\geq2)$.