@article{Zhang2024, 
author = {Chang-Xu Zhang and Fu-Tao Hu and Shu-Cheng Yang},
title = {On the (total) Roman domination in Latin square graphs},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {1},
pages = {594-606},
keywords = {Latin square, Latin square graphs, Roman domination number, total Roman domination number},
url = {https://www.sciopen.com/article/10.3934/math.2024031},
doi = {10.3934/math.2024031},
abstract = {Latin square, also known as Latin square matrix, refers to a kind of    n  ×  n matrix, in which there are exactly    n different symbols and each symbol appears exactly once in each row and column. A Latin square graph    Γ  (  L  ) is a simple graph associated with a Latin square    L. This paper studied the relationships between the (total) Roman domination number and (total) domination number of Latin square graph    Γ  (  L  ). We showed that        γ          R        (  Γ  (  L  )  )  =  2  γ  (  Γ  (  L  )  ) or        γ          R        (  Γ  (  L  )  )  =  2  γ  (  Γ  (  L  )  )  −  1, and        γ          t      R        (  Γ  (  L  )  )  ≥            8              γ        t            (      Γ      (      L      )      )        5   for    n  ≥  2. In 2021, Pahlavsay et al. proved    γ  (  Γ  (  L  )  )  ≥  ⌈      n    2    ⌉ and        γ    t    (  Γ  (  L  )  )  ≥  ⌈            4      n      −      2        7    ⌉ for    n  ≥  2. In this paper, we showed that        γ    R    (  Γ  (  L  )  )  ≥  2  ⌈      n    2    ⌉ (equality holds if, and only if,    γ  (  Γ  (  L  )  )  =  ⌈      n    2    ⌉) and        γ    t    (  Γ  (  L  )  )  >            4      n        7   for    n  ≥  2. Since        γ    R    (  G  )  ≤  2  γ  (  G  ) and        γ          t      R        (  G  )  ≤  2      γ    t    (  G  ) for any graph    G, our results can deduce or improve Pahlavsay et al.'s results. Moreover, we characterized these Latin squares for        γ    R    (  Γ  (  L  )  )  =  2  ⌈      n    2    ⌉, which is equal to    γ  (  Γ  (  L  )  )  =  ⌈      n    2    ⌉.}
}