Long cycles and spectral radii in planar graphs

TL;DR

This paper investigates the relationship between spectral radius conditions and the existence of long cycles in large planar graphs, extending classical cycle results by incorporating spectral graph theory.

Contribution

It establishes a spectral radius threshold that guarantees long cycles in large planar graphs without the 4-connectedness condition, identifying extremal graphs achieving this bound.

Findings

Spectral radius condition ensures long cycles in large planar graphs.

Identifies extremal graphs that meet the spectral threshold.

Extends classical cycle existence results to spectral conditions.

Abstract

There is a rich history of studying the existence of cycles in planar graphs. The famous Tutte theorem on the Hamilton cycle states that every 4-connected planar graph contains a Hamilton cycle. Later on, Thomassen (1983), Thomas and Yu (1994) and Sanders (1996) respectively proved that every 4-connected planar graph contains a cycle of length $n-1, n-2$ and $n-3$. Chen, Fan and Yu (2004) further conjectured that every 4-connected planar graph contains a cycle of length $\ell$ for $\ell\in\{n,n-1,\ldots,n-25\}$ and they verified that $\ell\in \{n-4, n-5, n-6\}$. When we remove the ``4-connected" condition, how to guarantee the existence of a long cycle in a planar graph? A natural question asks by adding a spectral radius condition: What is the smallest constant $C$ such that for sufficiently large $n$, every graph $G$ of order $n$ with spectral radius greater than $C$ contains a long cycle in a planar graph? In this paper, we give a stronger answer to the above question. Let $G$ be a planar graph with order $n\geq 1.8\times 10^{17}$ and $k\leq \lfloor\log_2(n-3)\rfloor-8$ be a non-negative integer, we show that if $\rho(G)\geq \rho(K_2\vee(P_{n-2k-4}\cup 2P_{k+1}))$ then $G$ contains a cycle of length $\ell$ for every $\ell\in \{n-k, n-k-1, \ldots, 3\}$ unless $G\cong K_2\vee(P_{n-2k-4}\cup 2P_{k+1})$.