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Tight Toughness Bounds for Fractional (k, n)-Critical Graphs with Large n

Published in: Quaestiones Mathematicae
Volume 48 , issue 10 , 2025 , pages: 1537–1558

DOI: 10.2989/16073606.2025.2534451

Author(s): Wei Gao School of Mathematics, Hohai University, China , Weifan Wang School of Mathematical Sciences, Zhejiang Normal University, China , Yaojun Chen School of Mathematics, Nanjing University, China

Keywords: Primary: 05C70 , Secondary: 05C72 , Graph , toughness , fractional k-factor , fractional (k , n)-critical graph

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Abstract

A graph G is called a fractional (k, n)-critical graph if G − V ′ admits a fractional k-factor for any V ′ ⊆ V (G) with |V′| = n. The main result in this paper states the following facts: 1) G is fractional (2, n)-critical if t(G) > and n ≥ 3; 2) G is fractional (2, 2)-critical if t(G) > ; 3) G is fractional (k, n)-critical if t(G) > with n ≥ k ≥ 3. Furthermore, the sharpness of the given toughness bounds are illustrated by counterexamples.

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