The $L(3, 2, 1)$-Labeling on Bipartite Graphs (Communications in Mathematical Research, 2021)
An $L(3, 2, 1)$-labeling of a graph $G$ is a function from the vertex set $V(G)$ to the set of all nonnegative integers such that $|f(u)−f(v)|≥3$ if $d_G(u, v)=1$, $|f(u)−f(v)|≥2$ if $d_G(u, v)=2$, and $|f(u)−f(v)|≥1$ if $d_G(u, v)=3$. The $L(3, 2, 1)$-labeling problem is to find the smallest number $λ_3(G)$ such that there exists an $L(3, 2, 1)$-labeling function with no label greater than it. This paper studiesthe problem for bipartite graphs. We obtain some bounds of $λ_3$ for bipartite graphsand its subclasses. Moreover, we provide a best possible condition for a tree $T$ suchthat $λ_3(T)$ attains the minimum value.
Keywords: channel assignment problems; $L(2; 1)$-labeling; $L(3; 2; bipartite graph; tree.