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Cycles of lengths 3 and n-1 in digraphs under a Bang-Jensen-Gutin-Li type conditon

Zan-Bo Zhang, Wenhao Wu, Weihua He

TL;DR

The paper investigates Bang-Jensen–Gutin–Li type degree conditions for digraphs and shows that, beyond guaranteeing Hamiltonicity, such conditions also ensure the existence of a $3$-cycle and an $(n-1)$-cycle in strong digraphs, with explicitly characterized exceptional cases. The authors prove these results via case analyses, leveraging structural lemmas and the multi-insertion technique, and discuss implications for pancyclicity, including conjectures that the condition may imply cycles of all lengths. The work situates itself within the broader framework of locally semicomplete digraphs and round decompositions, highlighting both the potential for full pancyclicity and the natural exceptional classes that arise. These contributions extend the applicability of BJG-Li type conditions from Hamiltonicity to sharper cycle-length guarantees, advancing understanding of cycle structure in directed graphs under degree-restriction hypotheses.

Abstract

Bang-Jensen-Gutin-Li type conditions are the conditions for hamiltonicity of digraphs which impose degree restrictions on nonadjacent vertices which have a common in-neighbor or a common out-neighbor. They can be viewed as an extension of Fan type conditions in undirected graphs, as well as generalization of locally (in-, out-)semicomplete digraphs. Since their first appearance in 1996, various Bang-Jensen-Gutin-Li type conditions for hamitonicity have come forth. In this paper we establish a condition of Bang-Jensen-Gutin-Li type which implies not only a hamiltonian cycle but also a 3-cycle and an (n-1)-cycle, with well-characterized exceptional graphs. We conjecture that this condition implies the existence of cycle of every length.

Cycles of lengths 3 and n-1 in digraphs under a Bang-Jensen-Gutin-Li type conditon

TL;DR

The paper investigates Bang-Jensen–Gutin–Li type degree conditions for digraphs and shows that, beyond guaranteeing Hamiltonicity, such conditions also ensure the existence of a -cycle and an -cycle in strong digraphs, with explicitly characterized exceptional cases. The authors prove these results via case analyses, leveraging structural lemmas and the multi-insertion technique, and discuss implications for pancyclicity, including conjectures that the condition may imply cycles of all lengths. The work situates itself within the broader framework of locally semicomplete digraphs and round decompositions, highlighting both the potential for full pancyclicity and the natural exceptional classes that arise. These contributions extend the applicability of BJG-Li type conditions from Hamiltonicity to sharper cycle-length guarantees, advancing understanding of cycle structure in directed graphs under degree-restriction hypotheses.

Abstract

Bang-Jensen-Gutin-Li type conditions are the conditions for hamiltonicity of digraphs which impose degree restrictions on nonadjacent vertices which have a common in-neighbor or a common out-neighbor. They can be viewed as an extension of Fan type conditions in undirected graphs, as well as generalization of locally (in-, out-)semicomplete digraphs. Since their first appearance in 1996, various Bang-Jensen-Gutin-Li type conditions for hamitonicity have come forth. In this paper we establish a condition of Bang-Jensen-Gutin-Li type which implies not only a hamiltonian cycle but also a 3-cycle and an (n-1)-cycle, with well-characterized exceptional graphs. We conjecture that this condition implies the existence of cycle of every length.
Paper Structure (3 sections, 10 theorems, 14 equations)

This paper contains 3 sections, 10 theorems, 14 equations.

Table of Contents

  1. Introduction and terminology
  2. Proof of Theorem \ref{['thm:BGLTriangle']}: the existence of a $\boldsymbol{3}$-cycle
  3. Proof of Theorem \ref{['thm:BGL_N-1_Cycle']}: the existence of an $\boldsymbol{(n-1)}$-cycle

Key Result

Theorem 1.1

(BJGL1996) Let $D$ be a strong digraph on $n\ge 2$ verties. Suppose that, for every nonadjacent dominated pair $\{x,y\}$, either $d(x) \ge n$ and $d(y) \ge n - 1$ or $d(x) \ge n - 1$ and $d(y) \ge n$. Then $D$ is hamiltonian.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • ...and 3 more