Odd covers for complete graphs and complete 3-graphs

Abstract

The Graham-Pollak theorem says that one needs at least $n - 1$ complete bipartite graphs to cover each edge of a complete graph $K_{n}$ on $n$ vertices exactly once. The odd cover problem is a parity analogue which seeks the minimum number of complete bipartite graphs, denoted by $b_2(n)$, such that each edge of $ K_n $ is covered an odd number of times. An odd cover of a complte 3-graph $K_n^{(3)}$ on $n$ vertices is a family of complete $3$-partite $3$-graphs such that every triple is covered an odd number of times. Let $b_3(n)$ be the minimum size of such a family. The values of $b_2(n)$ and $b_3(n)$ are determined for some $n$ in several previous works. In this paper, we first determine the value of $b_2(n)$ for all $n$, which confirms a conjecture due to Buchanan et al. (JGT, 2026), and then show $b_3(n+1)=b_2(n)$ by which the value of $b_3(n)$ is determined for all $n$, that resolves a question posed by Leader and Tan (EJC, 2026).