$L(d_1, d_2, \ldots, d_t)$-Number $\lambda(C_n; d_1, d_2, \ldots, d_t)$ of Cycles

Abstract:

      An $L(d_1, d_2, \ldots, d_t)$-labeling of a graph $G$ is a function $f$ from its vertex set $V(G)$ to the set $\{0, 1, \ldots, k\}$ for some positive integer $k$ such that $|f(x)-f(y)|\geq d_i$, if the distance between vertices $x$ and $y$ in $G$ is equal to $i$ for $i=1, 2, \ldots, t$. The $L(d_1, d_2, \ldots, d_t)$-number $\lambda(G; d_1, d_2, \ldots, d_t)$ of $G$ is the smallest integer number $k$ such that $G$ has an $L(d_1, d_2, \ldots, d_t)$-labeling with $\max\{f(x)| x\in V(G)\}=k$. In this paper, we obtain the exact values for $\lambda(C_n; 2, 2, 1)$ and $\lambda(C_n; 3, 2, 1)$, and present lower and upper bounds for $\lambda(C_n; 2,\ldots, 2, 1, \ldots, 1)$