A note on (s, t)-relaxed L(2, 1)-labeling of graphs

Let $$G=(V, E)$$ G = ( V , E ) be a graph. For two vertices u and v in G, we denote $$d_G(u, v)$$ d G ( u , v ) the distance between u and v. A vertex v is called an i-neighbor of u if $$d_G(u,v)=i$$ d G ( u , v ) = i . Let s, t and k be nonnegative integers. An (s, t)-relaxed k-L(2, 1)-labeling of a graph G is an assignment of labels from $$\{0, 1, \ldots , k\}$$ { 0 , 1 , … , k } to the vertices of G if the following three conditions are met: (1) adjacent vertices get different labels; (2) for any vertex u of G, there are at most s 1-neighbors of u receiving labels from $$\{f(u)-1,f(u)+1\}$$ { f ( u ) - 1 , f ( u ) + 1 } ; (3) for any vertex u of G, the number of 2-neighbors of u assigned the label f(u) is at most t. The (s, t)-relaxed L(2, 1)-labeling number $$\lambda _{2,1}^{s,t}(G)$$ λ 2 , 1 s , t ( G ) of G is the minimum k such that G admits an (s, t)-relaxed k-L(2, 1)-labeling. In this article, we refute Conjecture 4 and Conjecture 5 stated in (Lin in J Comb Optim. doi: 10.1007/s10878-014-9746-9 , 2013).