Abstract: A 3-path $ uvw $ in a 3-polytope is an $ (i,j,k) $-path if $ d(u)\leq i $, $ d(v)\leq j $, and $ d(w)\leq k $,where $ d(x) $ is the degree of a vertex $ x $.It is well known that each 3-polytope has a vertex of degree at most 5 called minor.A description of 3-paths in a 3-polytope is minor or major if the central item of its every triplet is at least 6.Back in 1922, Franklin proved that each 3-polytope with minimum degree 5 has a $ (6,5,6) $-path which description is tight.In 2016, we proved that each polytope with minimum degree 5 has a $ (5,6,6) $-path which is also tight.For arbitrary 3-polytopes, Jendrol’ (1996) gave the following description of3-paths:(10,3,10), (7,4,7),(6,5,6),(3,4,15),(3,6,11),(3,8,5),(3,10,3),(4,4,11),(4,5,7),(4,7,5),but it is unknown whether the description is tight or not.The first tight description of 3-paths was obtainedin 2013 by Borodin et al.:(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 tight description was given by Borodin, Ivanova, and Kostochka in 2017:(3,15,3), (3,10,4), (3,8,5), (4,7,4), (5,5,7), (6,5,6), (3,4,11), (4,4,9), (6,4,7)The purpose of this paper is to obtain the following major tight descriptions of 3-paths for arbitrary 3-polytopes:(3,18,3),(3,11,4),(3,8,5),(3,7,6),(4,9,4),(4,7,5),(5,6,6). © 2021, Pleiades Publishing, Ltd.

Cite: Borodin O.V. , Ivanova A.O.
A Tight Description of 3-Polytopes by Their Major 3-Paths
Siberian Mathematical Journal. 2021. V.62. N3. P.400-408. DOI: 10.1134/S0037446621030022 WOS Scopus OpenAlex

Identifiers:

Web of science:	WOS:000655743500002
Scopus:	2-s2.0-85107210005
OpenAlex:	W3171945737

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