The Splitting Number and Skewness of $C_n\times C_m$

Abstract

The skewness of a graph $G$ is the minimum number of edges that need to be deleted from $G$ to produce a planar graph. The splitting number of a graph $G$ is the minimum number of splitting steps needed to turn $G$ into a planar graph; where each step replaces some of the edges $\{u,v\}$ incident to a selected vertex $u$ by edges $\{u^{\prime },v\}$, where $u^{\prime }$ is a new vertex. We show that the splitting number of the toroidal grid graph $C_n\times C_m$ is $min\{n,m\}–2{\delta }_{n,3}{\delta }_{m,3}–{\delta }_{n,4}{\delta }_{m,3}–{\delta }_{n,3}{\delta }_{m,4}$ and its skewness is $min\{n,m\}–{\delta }_{n,3}{\delta }_{m,3}-{\delta }_{n,4}{\delta }_{m,3}–{\delta }_{n,3}{\delta }_{m,4}$. Here, $\delta$ is the Kronecker symbol, i.e., ${\delta }_{i,j}$ is $1$ if $i=j$, and $0$ if $i\ne j$.