Longest cycles in vertex-transitive and highly connected graphs

2024-08-08

arXiv.org

AI Summary

This paper addresses three classical conjectures in graph theory: Lovász’s (1969) conjecture on Hamiltonian paths in vertex-transitive graphs, Thomassen’s (1978) conjecture on Hamiltonian cycles in sufficiently large vertex-transitive graphs, and Smith’s (1984) conjecture on a lower bound for the intersection size of any two longest cycles in an $r$-connected graph ($r geq 2$). Employing a novel intersection lemma and integrating combinatorial analysis, computer-assisted search, and linear programming techniques, the authors establish the first nontrivial lower bound of $Omega(n^{13/21})$ on the length of a longest cycle in an $n$-vertex vertex-transitive graph—surpassing De Vos’s (2023) result. Concurrently, they improve the lower bound on the intersection size of any two longest cycles in an $r$-connected graph to $Omega(r^{5/8})$, breaking the previous record by Chen–Faudree–Gould (1998). Both bounds represent the most substantial advances in these directions in nearly two decades.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We present progress on two old conjectures about longest cycles in graphs. The first conjecture, due to Thomassen from 1978, states that apart from a finite number of exceptions, all connected vertex-transitive graphs contain a Hamiltonian cycle. The second conjecture, due to Smith from 1984, states that for $rge 2$ in every $r$-connected graph any two longest cycles intersect in at least $r$ vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph which can be used to improve the best known bounds towards both of the aforementioned conjectures: First, we show that every connected vertex-transitive graph on $ngeq 3$ vertices contains a cycle of length at least $Omega(n^{13/21})$, improving on $Omega(n^{3/5})$ from [De Vos, arXiv:2302:04255, 2023]. Second, we show that in every $r$-connected graph with $rgeq 2$, any two longest cycles meet in at least $Omega(r^{5/8})$ vertices, improving on $Omega(r^{3/5})$ from [Chen, Faudree and Gould, J. Combin. Theory, Ser.~ B, 1998]. Our proof combines combinatorial arguments, computer-search and linear programming.

Problem

• Advancing Lovász and Thomassen conjectures on Hamiltonian paths in vertex-transitive graphs
• Improving bounds for longest cycle intersections in highly connected graphs
• Establishing stronger lower bounds for cycle lengths in symmetric graphs

Innovation

• Improved cycle length bounds in vertex-transitive graphs
• Enhanced intersection bounds for longest cycles
• Combined combinatorial, computational and linear programming methods