# Self-complementary completions on six vertices

arXiv: https://arxiv.org/abs/2609.20231 · HTML: https://mlsanity.com/paper/2609.20231/

## Key facts

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- First posted on arXiv 27 Jul 2026.

## Abstract (arXiv)

Let (cthreshold(n)) be the largest integer (q) such that every loopless digraph on (n) vertices with at most (q) arcs is isomorphic to a spanning subdigraph of a self-complementary digraph of order (n). We prove that (cthreshold(6)=7). The upper bound is witnessed by [ bK3dunion (xlongrightarrow ylongrightarrow z), ] and follows from a direct argument with a self-complementing permutation. We also determine the complete eight-arc obstruction layer: it consists of five isomorphism classes, or three after converse digraphs are identified. All five are arc-minimal. Each nevertheless packs with an isomorphic copy of itself, so ordinary packing is strictly weaker than same-order self-complementary completion already at this first failure layer.