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An  L (2, 1)- coloring  (or  labeling ) of a graph  G  is a mappingsuch thatifandifThe  span of an L (2, 1) -coloring  is the maximum color assigned by it. The minimum of span of all possible  L (2, 1)-colorings of a graph  G  is called the  span  of  G  and is denoted byAn  L (2, 1)-coloring  f  of a graph  G  is called (i)  no-hole  if  f  uses all the colors from(ii)  irreducible  if no color of the vertices can be decreased and yield another  L (2, 1)-coloring of  G , (iii)  irreducible no-hole  (in short  inh-coloring ) if  f  is both irreducible and no-hole. The  inh-span  of an inh-colorable graph G, denoted byis defined asThe  lexicographic product  of two graphs  G  and  H  is the graphwith vertex setin which two vertices ( x ,  y ) andare adjacent precisely iforandIn this paper, we give improved upper bounds for span of lexicographic product of arbitrary graphs. We study inh-colorability ofandwhere  G  is an arbitrary graph and  T  is a tree different from a star and a path. We show that for an arbitrary graph  G1 , ifis inh-colorable then for any graph  G2  withandis inh-colorable and

L (2, 1)-coloring and irreducible no-hole coloring of lexicographic product of graphs