@article{Tian2024, 
author = {Shuangliang Tian and Ping Chen},
title = {Edge-coloring of generalized lexicographic product of graphs},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {6},
pages = {15988-15995},
keywords = {edge-coloring, decomposition, generalized lexicographic product},
url = {https://www.sciopen.com/article/10.3934/math.2024774},
doi = {10.3934/math.2024774},
abstract = {An edge-coloring of a graph    G is an assignment of colors to its edges so that no two edges incident to the same vertex receive the same color. The chromatic index of    G, denoted by        χ    ′    (  G  ), is the least    k for which    G has a    k edge-coloring. Graphs with        χ    ′    (  G  )  =  Δ  (  G  ) are said to be Class 1, and graphs with        χ    ′    (  G  )  =  Δ  (  G  )  +  1 are said to be Class 2. Let    G be a graph with    V  (  G  )  =  {      t    1    ,      t    2    ,  …  ,      t    n    },    n  ≥  2, and        h    n    =  (      H    i        )          i      ∈      {      1      ,      2      ,      …      ,      n      }       be a sequence of vertex-disjoint graphs with    V  (      H    i    )  =  {  (      t    i    ,      y    1    )  ,  (      t    i    ,      y    2    )  ,  …  ,  (      t    i    ,      y                  m        i              )  },        m    i    ≥  1. The generalized lexicographic product    G  [      h    n    ] of    G and        h    n    =  (      H    i        )          i      ∈      {      1      ,      2      ,      …      ,      n      }       is a simple graph with vertex set        ⋃          i      =      1              n        V  (      H    i    ), in which    (      t    i    ,      y    p    ) is adjacent to    (      t    j    ,      y    q    ) if and only if either        t    i    =      t    j   and    (      t    i    ,      y    p    )  (      t    i    ,      y    q    )  ∈  E  (      H    i    ) or        t    i        t    j    ∈  E  (  G  ). If    G is a complete graph with order 2, then    G  [      h    2    ] denotes a join        H    1    +      H    2   of vertex-disjoint graphs        H    1   and        H    2  . If        H    i    ≅  H for    i  =  1  ,  2  ,  …  ,  n, then    G  [      h    n    ]  =  G  [  H  ], where    G  [  H  ] denotes the lexicographic product of two graphs    G and    H. In this paper, we provide sufficient conditions for the generalized lexicographic product    G  [      h    n    ] of    G and        h    n    =  (      H    i        )          i      ∈      {      1      ,      2      ,      …      ,      n      }       to be Class 1, where all graphs in        h    n   have the same number of vertices.}
}