Paths of low weight in planar graphs

The existence of paths of low degree sum of their vertices in planar graphs is investigated. The main results of the paper are: 1. Every 3-connected simple planar graph G that contains a k-path, a path on k vertices, also contains a k-path P such that for its weight (the sum of degrees of its vertices) in G it holds ${\displaystyle w_G\left(P\right):={\sum }_{u\in V\left(P\right)}de{g}_G\left(u\right)\le \left(\frac{3}{2}\right)k²+\left(k\right)}$ 2. Every plane triangulation T that contains a k-path also contains a k-path P such that for its weight in T it holds ${\displaystyle w_T\left(P\right):={\sum }_{u\in V\left(P\right)}de{g}_T\left(u\right)\le k²+13k}$ 3. Let G be a 3-connected simple planar graph of circumference c(G). If c(G) ≥ σ| V(G)| for some constant σ > 0 then for any k, 1 ≤ k ≤ c(G), G contains a k-path P such that ${\displaystyle w_G\left(P\right)={\sum }_{u\in V\left(P\right)}de{g}_G\left(u\right)\le (\frac{3}{\sigma }+3)k}$.