Аннотация:
We consider plane graphs with large enough girth g, minimum degree δ at least 2 and no (k+1)-paths consisting of vertices of degree 2, where k⩾1. In 2016, Hudák, Maceková, Madaras, and Široczki studied the case k=1, which means that no two 2-vertices are adjacent, and proved, in particular, that there is a 3-vertex whose all three neighbors have degree 2 (called a soft 3-star), provided that g⩾10, which bound on g is sharp. For the first open case k=2 it was known that a soft 3-star exists if g⩾14 but may not exist if g⩽12. In this paper, we settle the case k=2 by presenting a construction with g=13 and no soft 3-star. For all k⩾3, we prove that soft 3-stars exist if g⩾4k+6 but, as follows from our construction, possibly not exist if g⩽3k+7. We conjecture that in fact soft 3-stars exist whenever g⩾3k+8.
The first author' work was supported by Mathematical Center in Akademgorodok under
agreement No. 075-15-2019-1613 with the Ministry of Science and Higher
Education of the Russian Federation. The second author' work was
supported by the Ministry of Science and Higher Education of the
Russian Federation (Grant No. FSRG-2020-0006).
Поступила4 сентября 2020 г., опубликована 18 ноября 2020 г.