 A Result on $K$-(2,1)-Total Choosability of Planar Graphs
Received:March 14, 2021  Revised:December 23, 2021
Key Word: $L$-(2,1)-total labeling   $k$-(2,1)-total choosable   planar graphs
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.12071265) and the Natural Science Foundation of Shandong Province (Grant No.ZR2019MA032).
 Author Name Affiliation Yan SONG Department of Mathematics, Shandong Normal University, Shandong 250014, P. R China Lei SUN Department of Mathematics, Shandong Normal University, Shandong 250014, P. R China
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A list assignment of a graph $G$ is a function $L:V(G)\cup E(G)\rightarrow 2^{N}$. A graph $G$ is $L$-(2,1)-Total labeling if there exists a function $c$ such that $c(x)\in L(x)$ for all $x\in V(G)\cup E(G)$, $|c(u)-c(v)|\geq 1$ if $uv\in E(G)$, $|c(e_{1})-c(e_{2})|\geq 1$ if the edges $e_{1}$ and $e_{2}$ are adjacent, and $|c(u)-c(e)|\geq 2$ if the vertex $u$ is incident to the edge $e$. A graph $G$ is $k$-(2,1)-Total choosable if G is $L$-(2,1)-Total labeling for every list assignment $L$ provided that $|L(x)|=k,x\in V(G)\cup E(G)$. The $(2,1)$-Total choice number of $G$, denoted by $C_{2,1}^{T}(G)$, is the minimum $k$ such that $G$ is $k$-(2,1)-Total choosable. In this paper, we prove that if $G$ is a planar graph with $\Delta(G)\geq 11$, then $C_{2,1}^{T}(G)\leq\Delta+4$.