Matchings and Independence in Bounded-Occupancy Token Graphs: Concurrent Independent Development

Abstract

A token configuration assigns nonnegative occupancies of total size k to the vertices of a graph; a move transfers one token along one edge. This article records concurrent independent development of a matching proof of the independence-number conjecture of Reyes, Dalfó and Fiol. An earlier public solution by Shivam Patel uses the same core construction for unrestricted occupancy. If a bipartite base graph has a matching saturating one part, the lifted matching saturates the odd-occupancy configurations, so the even configurations form a maximum independent set. We present this construction together with its bounded-occupancy consequences: even common capacities retain parity saturation, unequal capacities admit an explicit local classification, and odd common capacities allow counterexamples. Products of paths also give sharp general matching bounds, exact Shannon capacities in the saturated cases, and fixed-host asymptotic densities. Concurrent development and prior work. The mathematical statements and proofs are unchanged, and comparison with prior work has identified no mathematical error. Patel's complete solution was publicly posted on 21 August 2026, before this article's first public version on 6 September 2026. Both proofs use the same matched-edge-total and product-of-paths construction. This corrected version preserves an independently developed exposition and capacity consequences, without a first-solution priority claim. The proofs have not been formalized in a proof assistant. The article is retained as an archival exposition.