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

Abstract

Let $G=(V, E)$ be a graph. For two vertices u and v in G, we denote $d_G(u, v)$ the distance between u and v. A vertex v is called an i-neighbor of u if $d_G(u,v)=i$. 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 $\{0, 1, \ldots , k\}$ 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 $\{f(u)-1,f(u)+1\}$; (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 $\lambda _{2,1}^{s,t}(G)$ 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).