Long cycles and spectral radii in planar graphs

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 for ∈,n-1,…,n-25\ and they verified that ∈ -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≥ 1.8× 10¹⁷ and k≤ log₂(n-3)-8 be a non-negative integer, we show that if ρ(G)≥ ρ(K₂(Pₙ₋₂ₖ₋₄∪ 2Pₖ₊₁)) then G contains a cycle of length for every ∈ -k, n-k-1, …, 3\ unless G K₂(Pₙ₋₂ₖ₋₄∪ 2Pₖ₊₁).