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Weight of 3-paths in sparse plane graphs. (English) Zbl 1323.05058
Summary: We prove precise upper bounds for the minimum weight of a path on three vertices in several natural classes of plane graphs with minimum degree 2 and girth \(g\) from 5 to 7. In particular, we disprove a conjecture by S. Jendrol’ and M. Maceková [Discrete Math. 338, No. 2, 149–158 (2015; Zbl 1302.05040)] concerning the case \(g=5\) and prove the tightness of their upper bound for \(g=5\) when no vertex is adjacent to more than one vertex of degree 2. For \(g\geq8\), the upper bound recently found by Jendrol’ and Maceková [loc. cit.] is tight.

MSC:
05C22 Signed and weighted graphs
05C10 Planar graphs; geometric and topological aspects of graph theory
05C42 Density (toughness, etc.)
05C38 Paths and cycles
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