Crossing number of graphs and $\mathsf{\Delta Y}$-move

Abstract

The crossing number of a graph is the minimum number of double points over all generic immersions of the graph into the plane. In this paper we investigate the behavior of crossing number under a graph transformation, called $\mathsf{\Delta Y}$-move, on the complete graph $K_n$. Concretely it is shown that for any $k\in \mathbb{N}$, there exist a natural number $n$ and a sequence of $\mathsf{\Delta Y}$-moves $K_n\rightarrow G^{(1)}\rightarrow \cdots \rightarrow G^{(k)}$ which is decreasing with respect to the crossing number. We also discuss the decrease of crossing number for relatively small $n$.