@article{Cai2022, 
author = {Jin Cai and Shuangliang Tian and Lizhen Peng},
title = {On star and acyclic coloring of generalized lexicographic product of graphs},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {8},
pages = {14270-14281},
keywords = {star coloring, acyclic coloring, generalized lexicographic product, colorful neighbor star coloring, colorful neighbor acyclic coloring},
url = {https://www.sciopen.com/article/10.3934/math.2022786},
doi = {10.3934/math.2022786},
abstract = {A    s  t  a  r    c  o  l  o  r  i  n  g of a graph    G is a proper vertex coloring of    G such that any path of length 3 in    G is not bicolored. The    s  t  a  r    c  h  r  o  m  a  t  i  c    n  u  m  b  e  r        χ    s    (  G  ) of    G is the smallest integer    k for which    G admits a star coloring with    k colors. A    a  c  y  c  l  i  c    c  o  l  o  r  i  n  g of    G is a proper coloring of    G such that any cycle in    G is not bicolored. The    a  c  y  c  l  i  c    c  h  r  o  m  a  t  i  c    n  u  m  b  e  r of    G, denoted by    a  (  G  ), is the minimum number of colors needed to acyclically color    G. In this paper, we present upper bound for the star and acyclic chromatic numbers of the generalized lexicographic product    G  [      h    n    ] of graph    G and disjoint graph sequence        h    n  , where    G exists a    k  −colorful neighbor star coloring or    k  −colorful neighbor acyclic coloring. In addition, the upper bounds are tight.}
}