Dirac Graphs Without a Spanning Near-Square of an Odd Cycle: A Negative Answer to a Question of Heinig

At the 2014 Oberwolfach workshop on combinatorics, P. Heinig asked whether, for every odd n ≥ 7, every n-vertex graph with minimum degree at least ⌈n/2⌉ contains a spanning copy of the graph obtained from the square of an n-cycle by deleting every other edge on the periphery until exactly three consecutive vertices of degree 4 remain. A positive answer would have given a structural reason why the Hamilton cycles of such graphs generate their cycle space. We show that the answer is negative for every odd n ≥ 7, under both natural readings of "periphery", and for n = 9 under every reading. The counterexample is the complete bipartite graph K_{(n+1)/2,(n−1)/2} with a perfect matching, or a matching and one path with two edges, added inside the larger side; for n = 7 it is the graph that Heinig himself used as a positive example for the cycle-space question. The proof classifies the independent sets of size (n−1)/2 in Heinig's graph. The cycle-space question itself has since been settled for all large odd n by Hou and Yin, and it is not affected. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-12861-021. Paper page: https://eulersolve.org/papers/owr-12861-021/