@article{Pan2026, 
author = {Quanru Pan and Sizhong Zhou},
title = {Sufficient conditions for isolated tough graphs to have path-factors},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {5},
pages = {13371-13383},
keywords = {isolated toughness, size, distance signless Laplacian spectral radius, P≥2-factor},
url = {https://www.sciopen.com/article/10.3934/math.2026551},
doi = {10.3934/math.2026551},
abstract = {Let    G be a connected graph with    n vertices, where    n is a positive integer. The size of    G is denoted by    e  (  G  ). The isolated toughness of    G, denoted by    I  (  G  ), is defined by     I  (  G  )  =  min      {                                        |                    S                      |                                    i          (          G          −          S          )                    :      S      ⊆      V      (      G      )            and            i      (      G      −      S      )      ≥      2        }  or    I  (  G  )  =  +  ∞ if    G is complete. A graph    G is called isolated    r-tough if    I  (  G  )  ≥  r. The distance signless Laplacian matrix        Q    (  G  ) of    G is defined by        Q    (  G  )  =  T  r  (  G  )  +      D    (  G  ), where        D    (  G  ) denotes the distance matrix of    G and    T  r  (  G  ) is the diagonal matrix of the vertex transmissions in    G. The largest eigenvalue of        Q    (  G  ), denoted by    η  (  G  ), is called the distance signless Laplacian spectral radius of    G. A        P          ≥      k      -factor means a path factor with every component containing at least    k vertices, where    k is an integer with    k  ≥  2. In this paper, we aim to establish two tight sufficient conditions based on    e  (  G  ) and    η  (  G  ) to guarantee that a graph    G contains a        P          ≥      2      -factor. Let    G be a connected isolated        t          2      t      +      1      -tough graph of order    n, where    t  ≥  1 is an integer. Then the following two results hold.(ⅰ) If    n  ≥  6  t  +  2 and    e  (  G  )  ≥  e  (      K    t    ∨  (      K          n      −      3      t      −      1        ∪  (  2  t  +  1  )      K    1    )  ), then    G contains a        P          ≥      2      -factor unless    G  =      K    t    ∨  (      K          n      −      3      t      −      1        ∪  (  2  t  +  1  )      K    1    ).(ⅱ) If    n  ≥  9  t  +  2 and    η  (  G  )  ≤  η  (      K    t    ∨  (      K          n      −      3      t      −      1        ∪  (  2  t  +  1  )      K    1    )  ), then    G contains a        P          ≥      2      -factor unless    G  =      K    t    ∨  (      K          n      −      3      t      −      1        ∪  (  2  t  +  1  )      K    1    ).}
}