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Characterization of minimally [formula omitted]-tough, [formula omitted]-free graphs for [formula omitted].

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Title: Characterization of minimally [formula omitted]-tough, [formula omitted]-free graphs for [formula omitted].
Authors: Ma, Hui1 (AUTHOR), Hu, Xiaomin1 (AUTHOR), Yang, Weihua1 (AUTHOR) yangweihua@tyut.edu.cn
Source: Discrete Applied Mathematics. Dec2025, Vol. 377, p43-50. 8p.
Subject Terms: *GRAPH connectivity, *MATHEMATICAL functions
Abstract: A graph G is minimally t -tough if the toughness of G is exactly t and the removal of any edge decreases the toughness. Kriesell's conjecture, stating that every minimally 1-tough graph has a vertex of degree 2, is still open for general graphs. Katona and Varga generalized Kriesell's conjecture that every minimally t -tough graph has a vertex of degree ⌈ 2 t ⌉ for any positive rational number t. We have confirmed Kriesell's conjecture for 2 K 2 -free graphs by showing that every minimally 1-tough, 2 K 2 -free graph is C 4 or C 5. In this paper, we prove that for 1 < t ≤ 2 , every minimally t -tough, 2 K 2 -free graph on at least 14 vertices has a vertex of degree ⌈ 2 t ⌉ by characterizing the structure of these graphs. [ABSTRACT FROM AUTHOR]
Database: Academic Search Index
Description
Abstract:A graph G is minimally t -tough if the toughness of G is exactly t and the removal of any edge decreases the toughness. Kriesell's conjecture, stating that every minimally 1-tough graph has a vertex of degree 2, is still open for general graphs. Katona and Varga generalized Kriesell's conjecture that every minimally t -tough graph has a vertex of degree ⌈ 2 t ⌉ for any positive rational number t. We have confirmed Kriesell's conjecture for 2 K 2 -free graphs by showing that every minimally 1-tough, 2 K 2 -free graph is C 4 or C 5. In this paper, we prove that for 1 < t ≤ 2 , every minimally t -tough, 2 K 2 -free graph on at least 14 vertices has a vertex of degree ⌈ 2 t ⌉ by characterizing the structure of these graphs. [ABSTRACT FROM AUTHOR]
ISSN:0166218X
DOI:10.1016/j.dam.2025.06.045