Weight of 3-paths in sparse plane graphs (Q888584)

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 \textit{S. Jendrol'} and \textit{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.