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Volume 25, Issue 1
The $L(3, 2, 1)$-Labeling on Bipartite Graphs

Wanlian Yuan, Mingqing Zhai & Changhong Lü

Commun. Math. Res., 25 (2009), pp. 79-87.

Published online: 2021-06

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  • 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.

  • Keywords

channel assignment problems, $L(2, 1)$-labeling, $L(3, 2, 1)$-labeling, bipartite graph, tree.

  • AMS Subject Headings

68R10, 05C15

  • Copyright

COPYRIGHT: © Global Science Press

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@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} }
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 -

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.

Yuan , WanlianZhai , Mingqing and Lü , Changhong. (2021). The $L(3, 2, 1)$-Labeling on Bipartite Graphs. Communications in Mathematical Research . 25 (1). 79-87. doi:
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