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, n-1, , n-25\ and they verified that \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 1. 8 10^17 and k ₂ (n-3) -8 be a non-negative integer, we show that if (G) (K₂ (P₍-₂₊-₄ 2P₊+₁) ) then G contains a cycle of length for every \n-k, n-k-1, , 3\ unless G K₂ (P₍-₂₊-₄ 2P₊+₁).