Total list weighting of Cartesian product of graphs

Abstract

A proper total weighting of a graph $G$ is a mapping $\varphi$ that assigns a real number as the weight to each vertex and each edge of $G$ so that for any two adjacent vertices $u$ and $v$, ${\sum }_{e\in E\left(u\right)}\varphi \left(e\right)+\varphi \left(u\right)\ne {\sum }_{e\in E\left(v\right)}\varphi \left(e\right)+\varphi \left(v\right)$. A graph $G=(V,E)$ is called $(k,k^{\prime })$-choosable if the following is true: If each vertex $v$ is assigned a set $L\left(v\right)$ of $k$ real numbers, and each edge $e$ is assigned a set $L\left(e\right)$ of $k^{\prime }$ real numbers, then there is a proper total weighting $\varphi$ with $\varphi \left(z\right)\in L\left(z\right)$ for any $z\in V\cup E$. In this paper, we prove that if $G$ is the Cartesian product of a path and a cycle or the Cartesian product of two cycles, then $G$ is $(1,3)$-choosable and $(2,2)$-choosable. This improves and extends the known results which were proved by Wong et al. in 2012.