TY - JOUR
T1 - The $L(3, 2, 1)$-Labeling on Bipartite Graphs
AU - Yuan , Wanlian
AU - Zhai , Mingqing
AU - Lü , Changhong
JO - Communications in Mathematical Research 
VL - 1
SP - 79
EP - 87
PY - 2021
DA - 2021/06
SN - 25
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/cmr/19284.html
KW - channel assignment problems, $L(2, 1)$-labeling, $L(3, 2, 1)$-labeling, bipartite graph, tree.
AB - <p style="text-align: justify;">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.</p>