# Paths of low weight in planar graphs

• Volume: 28, Issue: 1, page 121-135
• ISSN: 2083-5892

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## Abstract

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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 ${w}_{G}\left(P\right):={\sum }_{u\in V\left(P\right)}de{g}_{G}\left(u\right)\le \left(3/2\right)k²+\left(k\right)$ 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 ${w}_{T}\left(P\right):={\sum }_{u\in V\left(P\right)}de{g}_{T}\left(u\right)\le k²+13k$ 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 ${w}_{G}\left(P\right)={\sum }_{u\in V\left(P\right)}de{g}_{G}\left(u\right)\le \left(3/\sigma +3\right)k$.

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