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

Let G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V, E)$$\end{document} be a graph. For two vertices u and v in G, we denote dG(u,v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_G(u, v)$$\end{document} the distance between u and v. A vertex v is called an i-neighbor of u if dG(u,v)=i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_G(u,v)=i$$\end{document}. 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,…,k}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{0, 1, \ldots , k\}$$\end{document} 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}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{f(u)-1,f(u)+1\}$$\end{document}; (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 λ2,1s,t(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{2,1}^{s,t}(G)$$\end{document} 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).