Discussiones Mathematicae Graph Theory 28(1) (2008) 121-135
doi: 10.7151/dmgt.1396

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PATHS OF LOW WEIGHT IN PLANAR GRAPHS

Igor Fabrici1, Jochen Harant2  and  Stanislav Jendrol'1

1Institute of Mathematics
P.J. Safárik University
Jesenná 5, SK-04154 Košice, Slovak Republic
e-mail: stanislav.jendrol@upjs.sk,  igor.fabrici@upjs.sk

2Institute of Mathematics
Ilmenau Technical University
PF 10 05 65, D-98684 Ilmenau, Germany
e-mail: harant@mathematik.tu-ilmenau.de

Abstract

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

wG(P): =

u ∈ V(P) 
degG(u) ≤ 3
2
 k2+O(k) .

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

wT(P): =

u ∈ V(P) 
degT(u) ≤ 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

wG(P) =

u ∈ V(P) 
degG(u) ≤
3
σ
+3
k .
Keywords: planar graphs, polytopal graphs, paths, weight of an edge, weight of a path.

2000 Mathematics Subject Classification: 05C10, 05C38, 52B10.

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Received 22 November 2006
Revised 4 May 2007
Accepted 4 May 2007