On $d$-panconnected tournaments with large semidegrees

We prove the following new results. (a) Let $T$ be a regular tournament of order $2n+1\geq 11$ and $S$ a subset of $V(T)$. Suppose that $|S|\leq \frac{1}{2}(n-2)$ and $x$, $y$ are distinct vertices in $V(T)\setminus S$. If the subtournament $T-S$ contains an $(x,y)$-path of length $r$, where $3\leq r\leq |V(T)\setminus S|-2$, then $T-S$ also contains an $(x,y)$-path of length $r+1$. (b) Let $T$ be an $m$-irregular tournament of order $p$, i.e., $|d^+(x)-d^-(x)|\le m$ for every vertex $x$ of $T.$ If $m\leq \frac{1}{3}(p-5)$ (respectively, $m\leq \frac{1}{5}(p-3)$), then for every pair of vertices $x$ and $y$, $T$ has an $(x,y)$-path of any length $k$, $4\leq k\leq p-1$ (respectively, $3\leq k\leq p-1$ or $T$ belongs to a family $\cal G$ of tournaments, which is defined in the paper). In other words, (b) means that if the semidegrees of every vertex of a tournament $T$ of order $p$ are between $\frac{1}{3}(p+1)$ and $\frac{2}{3}(p-2)$ (respectively, between $\frac{1}{5}(2p-1)$ and $\frac{1}{5}(3p-4)$), then the claims in (b) hold. Our results improve in a sense related results of Alspach (1967), Jacobsen (1972), Alspach et al. (1974), Thomassen (1978) and Darbinyan (1977, 1978, 1979), and are sharp in a sense.