Irreducible No-Hole L(2, 1)-Coloring of Edge-Multiplicity-Paths-Replacement Graph

An (2, 1)- (or ) of a simple connected graph is a mapping : ( ) → ∪ {0} such that | )− )| ≥ 2 for all edges of , and | ) − )| ≥ 1 if and are at distance two in . The (2, 1)- , denoted by span( ), of is max{ ) : ∈ ( )}. The , denoted by λ( ), is the minimum span of all possible (2, 1)-colorings of . For an (2, 1)-coloring of a graph with span , an integer is a in if ∈ (0, ) and there is no vertex in such that ) = . An (2, 1)-coloring is a if there is no hole in it, and is an if color of none of the vertices in the graph can be decreased and yield another (2, 1)-coloring of the same graph. An , in short , of is an (2, 1)-coloring of which is both irreducible and no-hole. For an inh-colorable graph , the of , denoted by λ ), is defined as λ ) = min{span( ) : is an inh-coloring of . Given a function : ) → ℕ − {1}, and a positive integer ≥ 2, the ) of is the graph obtained by replacing every edge of with paths of length ) each. In this paper we show that ) is inh-colorable except possibly the cases ) ≥ 2 with equality for at least one but not for all edges and (i) Δ( ) = 2, = 2 or (ii) Δ ( ) ≥ 3, 2 ≤ ≤ 4. We find the exact value of λ )) in several cases and give upper bounds of the same in the remaining. Moreover, we find the value of λ( )) in most of the cases which were left by Lü and Sun in [ (2, 1)- , J. Comb. Optim. 31 (2016) 396–404].