@Article{CMR-25-79, author = {Yuan , WanlianZhai , Mingqing and Lü , Changhong}, title = {The $L(3, 2, 1)$-Labeling on Bipartite Graphs}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {25}, number = {1}, pages = {79--87}, abstract = {

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.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19284.html} }