Circuit orderability of cographic matroids of complete and complete bipartite graphs

We classify the circuit-orderable cographic matroids associated with two complete graph families. For every integer n >= 4, M*(K_n) is orderable if and only if n = 4 or n = 6. For all positive integers r,s, M*(K_{r,s}) is orderable if and only if min(r,s) <= 2. The main argument reconstructs a closed simplicial surface from an arbitrary consistent ordering, then uses a mod-two face equation and a link-degree obstruction. The positive K6 case is from an earlier projective-plane construction. The complete bipartite classification combines known results with a three-neighbor obstruction for K_{3,s}. These classifications concern the two specified graph families, not all cographic matroids.

Status: Internally checked preprint; external mathematical review and formal peer review are pending.

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