Find Subtrees of Specified Weight and Cycles of Specified Length in Linear Time

We apply the Euler tour technique to find subtrees of specified weight as follows. Let $k, g, N1, N2 \in \mathbb{N}$ such that $1 \leq k \leq N2$, $g + h > 2$ and $2k - 4g - h + 3 \leq N2 \leq 2k + g + h - 2$, where $h := 2N1 - N2$. Let $T$ be a tree of $N1$ vertices and let $c : V(T) \rightarrow \mathbb{N}$ be vertex weights such that $c(T) := \sum{v \in V(T)} c(v) = N_2$ and $c(v) \leq k$ for all $v \in V(T)$. We prove that a subtree $S$ of $T$ of weight $k - g + 1 \leq c(S) \leq k$ exists and can be found in linear time. We apply it to show, among others, the following: (i) Every planar hamiltonian graph $G = (V(G), E(G))$ with minimum degree $\delta \geq 4$ has a cycle of length $k$ for every $k \in {\lfloor \frac{|V(G)|}{2} \rfloor, \dots, \lceil \frac{|V(G)|}{2} \rceil + 3}$ with $3 \leq k \leq |V(G)|$. (ii) Every $3$-connected planar hamiltonian graph $G$ with $\delta \geq 4$ and $|V(G)| \geq 8$ even has a cycle of length $\frac{|V(G)|}{2} - 1$ or $\frac{|V(G)|}{2} - 2$. Each of these cycles can be found in linear time if a Hamilton cycle of the graph is given. This work was partially motivated by conjectures of Bondy and Malkevitch on cycle spectra of 4-connected planar graphs.