Tight Leading-Term Bounds for the ℤ₂-Genus of Complete Graphs

Abstract

A drawing of a graph on a surface is independently even if every pair of independent edges in the drawing crosses an even number of times. The Euler ℤ₂-genus eg₀(G) and the orientable ℤ₂-genus g₀(G) are the minimal Euler genus and orientable genus on which the graph G has an independently even drawing.
We prove tight leading-term bounds for eg₀(K_n) and g₀(K_n). For every n ≥ 3, we show that eg₀(K_n) ≥ ⌈((n-3)(n-4) - (2n-5))/6⌉ and g₀(K_n) ≥ ⌈((n-3)(n-4) - (2n-5))/12⌉ .
Together with the classical genus formulas for K_n, these bounds determine the Euler and orientable ℤ₂-genera of complete graphs up to an additive O(n) term. Thus, for complete graphs, passing from embeddings to independently even drawings does not reduce the leading term of the required genus.
The proof adapts a matrix-rank framework previously used for complete bipartite graphs. Starting from an independently even drawing of K_n on a surface of minimum ℤ₂-genus, we planarize the drawing and encode crossing parities in a matrix over ℤ₂. Parity constraints specific to complete graphs allow us to transform this matrix into a structured form whose rank we can lower bound in two different ways, together yielding the stated genus lower bounds.