A Counterexample to Wormald's Conjecture

Abstract

We give a counterexample to Wormald's conjecture. A 16-vertex cubic graph, the disjoint union of $K_{3,3}$ with a 10-vertex bridged graph built from two subdivided copies of $K_4$, has no partition of its edges into two isomorphic spanning linear forests. The obstruction is parity: any such partition would force every monochromatic component to have even order while requiring a five-vertex side of a bridge to be partitioned into such components. Adjoining copies of $K_4$ gives counterexamples in every order $16+4t$. The counterexample is disconnected and leaves the connected case unresolved. The full family is verified in Lean without custom axioms.