Abstract   A  k -coloring  c  of a graph   G  =  (  V  ,  E  )   is a  b-coloring  if for every color class     c    i    ,   1  ≤  i  ≤  k  , there is a vertex colored  i  whose neighborhood intersects every other color class     c    j     of  c . The  b -chromatic number of   G  ,    χ    b    (  G  )  , is the greatest  k  such that  G  admits a  b -coloring with  k  colors. Every optimal coloring of  G  is a  b -coloring. Therefore   χ  (  G  )  ≤    χ    b    (  G  )  .  G  is  b -continuous if for every   k  ,  χ  (  G  )  ≤  k  ≤    χ    b    (  G  )  ,  G   admits a  b -coloring with  k  colors. In this paper, we are interested in  b -continuous graphs   G  [  H  ]   which are the lexicographic product of two  b -continuous graphs  G  and  H . We give partial results on the spectrum of   G  [  H  ]   and we examine its  b -continuity for specific classes of  G  and  H .
['Cláudia Linhares Sales', 'Rafael Vargas', 'Leonardo Sampaio']
b-continuity and the lexicographic product of graphs☆