Light 3-Paths in 3-Polytopes without Adjacent Triangles

Abstract

Let $w_{k}$ be the maximum of the minimum degree-sum (weight) of vertices in $k$-vertex paths ($k$-paths) in 3-polytopes. Trivially, each 3-polytope has a vertex of degree at most 5, and so $w_{1}\leq 5$. Back in 1955, Kotzig proved that $w_{2}\leq 13$ (so there is an edge of weight at most 13), which is sharp. In 1993, Ando, Iwasaki, and Kaneko proved that $w_{3}\leq 21$, which is also sharp due to a construction by Jendrol’ of 1997. In 1997, Borodin refined this by proving that $w_{3}\leq 18$ for 3-polytopes with $w_{2}\geq 7$, while $w_{3}\leq 17$ holds for 3-polytopes with $w_{2}\geq 8$, where the sharpness of 18 was confirmed by Borodin et al. in 2013, and that of 17 was known long ago. Over the last three decades, much research has been devoted to structural and coloring problems on the plane graphs that are sparse in this or that sense. In this paper we deal with 3-polytopes without adjacent 3-cycles that is without chordal 4-cycle (in other words, without $K_{4}-e$). It is known that such 3-polytopes satisfy $w_{1}\leq 4$; and, moreover, $w_{2}\leq 9$ holds, where both bounds are sharp (Borodin, 1992). We prove now that each 3-polytope without chordal 4-cycles has a 3-path of weight at most 15; and so $w_{3}\leq 15$, which is sharp.