Cycles in 3-connected claw-free planar graphs and 4-connected planar graphs without 4-cycles

The cycle spectrum $\mathrm{CS}\left(G\right)$ of a graph $G$ is the set of the cycle lengths in $G$. Let $G$ be a graph class. For any integer $k\ge 3$, define $f_G\left(k\right)$ to be the least integer $k^{\prime }\ge k$ such that for any $G\in G$ with circumference at least $k,\mathrm{CS}\left(G\right)\cap \left[k,k^{\prime }\right]\ne \varnothing$. Denote by $P,P3,\mathrm{PC}$, and $\mathrm{PC}3$ the classes of 3-connected planar graphs, 3-connected cubic planar graphs, 3-connected claw-free planar graphs, and 3-connected claw-free cubic planar graphs, respectively. The values of $f_P\left(k\right)$ and $f_{P3}\left(k\right)$ were known for all $k\ge 3$. In the first part of this article, we prove the claw-free version of these results by giving the values of $f_{\mathrm{PC}}\left(k\right)$ and $f_{\mathrm{PC}3}\left(k\right)$ for all $k\ge 3$. In the second part we study the cycle spectra of 4-connected planar graphs without 4-cycles. Bondy conjectured that every 4-connected planar graph $G$ has all cycle lengths from 3 to $\mid V\left(G\right)\mid$ except possibly one even length. The truth of this conjecture would imply that every 4-connected planar graph $G$ without 4-cycles has all cycle lengths other than 4. It was already known that $\left\{3,5,\dots ,9\right\}\cup \left\{\lfloor \mid V\left(G\right)\mid ∕2\rfloor ,\dots ,\lceil \mid V\left(G\right)\mid ∕2\rceil +3\right\}\cup \left\{\mid V\left(G\right)\mid -7,\dots ,\mid V\left(G\right)\mid \right\}\subseteq \mathrm{CS}\left(G\right)$. We prove that $G$ can be embedded in the plane such that for any $k\in \left\{\lfloor \mid V\left(G\right)\mid ∕2\rfloor ,\dots ,\mid V\left(G\right)\mid \right\},G$ has a $k$-cycle $C$ and all vertices not in $C$ lie in the exterior of $C$.