Tight Descriptions of 3‐Paths in Normal Plane Maps

Abstract

We prove that every normal plane map (NPM) has a path on three vertices (3‐path) whose degree sequence is bounded from above by one of the following triplets: (3, 3, ∞), (3,15,3), (3,10,4), (3,8,5), (4,7,4), (5,5,7), (6,5,6), (3,4,11), (4,4,9), and (6,4,7). This description is tight in the sense that no its parameter can be improved and no term dropped. We also pose a problem of describing all tight descriptions of 3‐paths in NPMs and make a modest contribution to it by showing that there exist precisely three one‐term descriptions: (5, ∞, 6), (5, 10, ∞), and (10, 5, ∞).

All tight descriptions of 3-paths centered at 2-vertices in plane graphs with girth at least 6

“…In 2013, Borodin et al [11] gave the first tight description of 3-paths: every G with δ ≥ 3 and g ≥ 3 has a 3-path of one of the following types: (3,4,11), (3,7,5), (3,10,4), (3,15,3), (4,4,9), (6,4,8), (7,4,7), (6,5,6). Another similar tight description for δ ≥ 3 and g ≥ 3 was given by Borodin, Ivanova, and Kostochka [12].…”

Light subgraphs in graphs with average degree at most four

“…In [6], Borodin et al introduced an irreducible optimal unavoidable set of 3-paths. Formally, for the set of types of 3-paths A = {(x − 1 , y − 1 , z − 1 ), (x − 2 , y − 2 , z − 2 ), .…”

“…Theorem 2. 6. If G is a graph with δ(G) ≥ 2 and average degree less than 10 3 , then G has one of the following configurations: a (2, 2, ∞)-path, a (2, 3, 6 − )-path, a (3, 3, 3)-path, a (2, 4, 3 − )-path and a (2, 9 − , 2)-path.…”

Точное Описание $3$-Многогранников Их Старшими $3$-Цепями

“…Теорема Франклина [2] является фундаментальной в структурной теории плоских графов; она была обобщена или уточнена в нескольких направлениях, см., например, [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] и обзоры Йендроля и Фосса [19] и О. В. Бородина и А. О. Ивановой [20].…”

“…Поставленная в [12] проблема нахождения всех точных описаний 3-цепей в P 3 по-прежнему широко открыта; в частности, до сих пор не найдено ни одного точного младшего описания.…”