Crossing number of graphs and Δ Y-move

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 Δ Y-move, on the complete graph Kₙ. Concretely it is shown that for any k∈ N, there exist a natural number n and a sequence of Δ Y-moves Kₙ→ G⁽¹⁾→ ⋯ → G⁽ᵏ⁾ which is decreasing with respect to the crossing number. We also discuss the decrease of crossing number for relatively small n.