Abstract: Over the last decades, a lot of research has been devoted to structural and coloring problems on plane graphs that are sparse in this or that sense. In this note we deal with the densest among sparse 3-polytopes, namely those having no adjacent 3-cycles. Borodin (1996) proved that such 3-polytopes have a vertex of degree at most 4 and, moreover, an edge with the degree-sum of its end-vertices at most 9, where both bounds are sharp. By d(v)denote the degree of a vertex v. An edge e =xyin a 3-polytope is an (i, j)-edge if d(x) ≤iand d(y) ≤j. The well-known (3,5;4,4)-Archimedean solid corresponds to a plane quadrangulation in which every edge joins a 3-vertex with a 5-vertex. We prove that every 3-polytope with neither adjacent 3-cycles nor (3, 5)-edges has a 3-face with the degree-sum of its incident vertices (weight) at most 16, which bound is sharp.

Cite: Borodin O.V. , Ivanova A.O.
Light 3-faces in 3-polytopes without adjacent triangles
Discrete Mathematics. 2025. V.348. N1. 114299 :1-4. DOI: 10.1016/j.disc.2024.114299 WOS Scopus OpenAlex

Dates:

Submitted:	Dec 8, 2023
Accepted:	Oct 17, 2024
Published print:	Oct 25, 2024
Published online:	Oct 25, 2024

Identifiers:

Web of science:	WOS:001344032900001
Scopus:	2-s2.0-85206981992
OpenAlex:	W4403729798

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DB	Citing
OpenAlex	1
Scopus	1
Web of science	1

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