What question did this study set out to answer?

The aim is to determine conditions that classify a graph as almost 1-tough and the presence of F-factors.

April 17, 2026Open Access

A note on the almost 1-tough graph

Key Points

The aim is to determine conditions that classify a graph as almost 1-tough and the presence of F-factors.
Defined criteria for almost 1-tough graphs based on components and subsets of vertices.
Developed a set-valued function to categorize vertex connections in the graph.
Established lower bounds related to size and spectral radius for identifying almost 1-tough graphs.
Provided a sufficient condition for the existence of F-factors under specific constraints.
Showed that if the number of vertices assigned to set {1} is even, an F-factor can exist.

Abstract

A graph G is almost 1-tough if c (G-S)? |S| + 1 for any S? V (G), where c (G-S) is the number of components of G-S. Let F: V (G)? {1, 0, 2} be a set-valued function and F^-1 (1): = v? V (G):? (v) = {1}. A spanning subgraph H of G is called an F-factor if dH (v)? F (v) for all v? V (G). It is interesting to know whether a graph is almost 1-tough or a graph has an F-factor. In this note, we establish a lower bound on the size (resp. the spectral radius) to ensure a graph to be almost 1-tough. This also provide a sufficient condition for the existence of an F-factor for which every F: V (G)? {1, 0, 2} with | F^-1 (1) | even in a graph.

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Cite This Study

Zheng et al. (Wed,) studied this question.

synapsesocial.com/papers/69e1cefb5cdc762e9d857eac https://doi.org/https://doi.org/10.2298/fil2527569z

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