$L(2,1)$-Labeling of Hamiltonian graphs with Maximum Degree 3

An $L(2,1)$-labeling of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $d(x,y)=1$ and $|f(x)-f(y)|\ge 1$ if $d(x,y)=2$. The $L(2,1)$-labeling number $\lambda(G)$ of $G$ is ...

Let $$G=(V, E)$$G=(V,E) be a graph. Denote $$d_G(u, v)$$dG(u,v) the distance between two vertices $$u$$u and $$v$$v in $$G$$G. An $$L(2, 1)$$L(2,1)-labeling of $$G$$G is a function $$f: V \rightarrow \{0,1,\cdots \}$$f:V {0,1, } such that for any two ...

An $L(p,1)$-labeling of a graph is a function $f$ from the vertex set to the positive integers such that $|f(x)-f(y)|\geqslant p$ if dist$(x,y)=1$ and $|f(x)-f(y)|\geqslant 1$ if dist$(x,y)=2$, where dist$(x,y)$ is the distance between the two vertices $...