A Resolution of the McCarty Conjecture

Anant Godbole, Lybitina Koene, Grant Shirley

Published 2025-03-29, updated 2025-05-06Version 2

The McCarty Conjecture states that any McCarty Matrix (an $n\times n$ matrix $A$ with positive integer entries and each of the $2n$ row and column sums equal to $n$), can be additively decomposed into two other matrices, $B$ and $C$, such that $B$ has row and column sumsets both equal to $\{1, 2,... n\}$, and $C$ has row and column sumsets both equal to $\{0, 1,... n-1\}$. The problem can also be formulated in terms of bipartite graphs. In this paper we use probabilistic methods to resolve this conjecture.