Tight toughness, isolated toughness and binding number bounds for (𝒜, n)-critical deleted graphs

¹ School of Mathematics and Statistics, Qinghai Normal University, Xining 810001, P.R. China
² Academy of Plateau Science and Sustainability, People–s Government of Qinghai Province and Beijing Normal University, Xining, Qinghai Province

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 8 February 2025
Accepted: 22 December 2025

Abstract

For a family 𝒜 of connected graphs, an 𝒜-factor is a spanning subgraph of a graph, whose components are isomorphic to a member of 𝒜. A graph G is called an 𝒜-factor deleted graph if G – e has an 𝒜-factor for any e ∈ E(G). A graph G is called an (𝒜,n)-critical deleted graph if for any Vˊ ⊆ V(G) with |Vˊ| = n, G – Vˊ is an 𝒜-factor deleted graph. Clearly, an (𝒜,0)-critical deleted graph is an 𝒜-factor deleted graph. In this paper, we write 𝒜 = {K₂, C_2i+1: i ≥ 2} and give several sufficient conditions for a graph to be an (𝒜, n)-critical deleted graph.