(untitled) (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 studies the problem for bipartite graphs. We obtain some bounds of λ_3 for bipartite graphs and its subclasses. Moreover, we provide a best possible condition for a tree T such that λ_3(T) attains the minimum value.