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Path Extendable Tournaments

Zan-Bo Zhang, Weihua He, Hajo Broersma, Xiaoyan Zhang

TL;DR

This work investigates path extendability in tournaments through two central metrics, $\pi_2(T)$ and $i(T)$. It introduces the surplus of vertex pairs to bound the total number of $2$-paths and proves sharp upper bounds on $\pi_2(T)$, with equality characterized by (almost) doubly regular tournaments. The authors provide two main sufficient conditions for path extendability, including a combined $i(T)$–$\pi_2(T)$ bound, and show that higher $\pi_2(T)$ alone suffices in a stronger regime. They also establish path extendability results for regular tournaments and analyze random tournaments, proving that almost surely path extendability holds in the probabilistic model. Overall, the paper reveals that path extendability, while analogous to certain undirected Hamiltonian properties, demands notably stronger structural constraints and offers precise thresholds and probabilistic guarantees.

Abstract

A digraph $D$ is called \emph{path extendable} if for every nonhamiltonian (directed) path $P$ in $D$, there exists another path $P^\prime$ with the same initial and terminal vertices as $P$, and $V(P^\prime) = V (P)\cup \{w\}$ for a vertex $w \in V(D)\setminus V(P)$. Hence, path extendability implies paths of continuous lengths between every vertex pair. In earlier works of C. Thomassen and K. Zhang, it was shown that the condition of small $i(T)$ or positive $π_2(T)$ implies paths of continuous lengths between every vertex pair in a tournament $T$, where $i(T)$ is the irregularity of $T$ and $π_2(T)$ denotes for the minimum number of paths of length $2$ from $u$ to $v$ among all vertex pairs $\{u,v\}$. Motivated by these results, we study sufficient conditions in terms of $i(T)$ and $π_2(T)$ that guarantee a tournament $T$ is path extendable. We prove that (1) a tournament $T$ is path extendable if $i(T)< 2π_2(T)-(|T|+8)/6$, and (2) a tournament $T$ is path extendable if $π_2(T) > (7|T|-10)/36$. As an application, we deduce that almost all random tournaments are path extendable.

Path Extendable Tournaments

TL;DR

This work investigates path extendability in tournaments through two central metrics, and . It introduces the surplus of vertex pairs to bound the total number of -paths and proves sharp upper bounds on , with equality characterized by (almost) doubly regular tournaments. The authors provide two main sufficient conditions for path extendability, including a combined – bound, and show that higher alone suffices in a stronger regime. They also establish path extendability results for regular tournaments and analyze random tournaments, proving that almost surely path extendability holds in the probabilistic model. Overall, the paper reveals that path extendability, while analogous to certain undirected Hamiltonian properties, demands notably stronger structural constraints and offers precise thresholds and probabilistic guarantees.

Abstract

A digraph is called \emph{path extendable} if for every nonhamiltonian (directed) path in , there exists another path with the same initial and terminal vertices as , and for a vertex . Hence, path extendability implies paths of continuous lengths between every vertex pair. In earlier works of C. Thomassen and K. Zhang, it was shown that the condition of small or positive implies paths of continuous lengths between every vertex pair in a tournament , where is the irregularity of and denotes for the minimum number of paths of length from to among all vertex pairs . Motivated by these results, we study sufficient conditions in terms of and that guarantee a tournament is path extendable. We prove that (1) a tournament is path extendable if , and (2) a tournament is path extendable if . As an application, we deduce that almost all random tournaments are path extendable.
Paper Structure (12 sections, 18 theorems, 76 equations, 4 figures)

This paper contains 12 sections, 18 theorems, 76 equations, 4 figures.

Table of Contents

  1. Introduction, terminology and notation
  2. Main results
  3. Surplus and supremum of $\boldsymbol{\pi_2(T)}$; proof of Theorem \ref{['thm:pi_2Supremum']}
  4. Regular Tournaments; Proof of Theorem \ref{['thm:2PathsPathExt']}
  5. Condition on $\boldsymbol{i(T)}$ and $\boldsymbol{\pi_2(T)}$; Proof of Theorem \ref{['thm:pi_2(T)i(T)']} and Theorem \ref{['thm:pi_2(T)only']}
  6. Proof of Theorem \ref{['thm:pi_2(T)i(T)']}
  7. Proof of Lemma \ref{['lem:i(T)AndPi(T)']}
  8. Proof of Theorem \ref{['thm:pi_2(T)only']}
  9. Random Tournament; Proof of Theorem \ref{['thm:RandomTPi_2']} and Theorem \ref{['thm:RandomTPathExt']}
  10. Proof of Theorem \ref{['thm:RandomTPi_2']}
  11. Proof of Theorem \ref{['thm:RandomTPathExt']}
  12. Conclusion and open problems

Key Result

Theorem 1.1

(Thomassen Thomassen1980) If $T$ is a tournament on $n \geq 5k + 21$ vertices with $i(T) \leq k$, then $T$ is strongly panconnected.

Figures (4)

  • Figure 1: The digraph $D_{1}$ (left) and the tournaments in $\mathcal{T}_1$
  • Figure 2: The tournaments in $\mathcal{T}_2$
  • Figure 3: The tournament $T_0$
  • Figure 4: A regular tournament without $(u_0,u_1)$-$2$-paths

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 2.6
  • Theorem 2.7
  • ...and 15 more