On $d$-panconnected tournaments with large semidegrees

TL;DR

This paper establishes new results on the existence of long paths in large semidegree tournaments, improving previous bounds and providing sharp conditions for $d$-panconnectedness in regular and irregular tournaments.

Contribution

It introduces novel conditions on semidegrees that guarantee the existence of paths of all lengths, extending and sharpening earlier results in tournament path theory.

Findings

Regular tournaments of order ≥11 have extended path length properties.

Irregular tournaments with bounded semidegree differences contain paths of all lengths within certain bounds.

Results improve and are sharp relative to classical theorems by Alspach, Jacobsen, Thomassen, and Darbinyan.

Abstract

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.