Combinatorics and Graph Theory: Mistake Clinic

Module-Specific Mistake Radar

Error	Revealing example	Repair
Multiplication called statistical independence	Chair/deputy choices without replacement	Count extensions for every partial choice
Divide by symmetry without uniform fibers	Four-bit necklaces	List orbit sizes before division
Missing empty object	Empty word, empty coin multiset, empty function	State the counted set at size zero
Same recurrence means same sequence	No-11 words versus 1/2 tilings	Keep initial values and index shifts
OGF interpreted as only unordered objects	Ordered compositions	Derive the product/sequence construction
Walk confused with simple path	0–1–0	State permitted repetitions
Edge count alone used for tree	Triangle plus isolate	Require connectivity or acyclicity too
All non-bridges discarded	A triangle's spanning tree	Distinguish original bridges from chosen deletions
Even degrees suffice for Euler circuit	Two disjoint triangles	Check the edge-bearing component
Maximum confused with maximal	Greedy staff assignment	Supply an upper bound matching the witness size
Sparse implies planar	Subdivided K₅	Use a valid obstruction or embedding

Practice Mistake Checks

Check counted objects and labels in counting practice, state/base coverage in recurrence practice, and theorem hypotheses in graph practice. Use code fault experiments to connect a definition to a reproducible failure. A returned answer without a proof or witness is not enough.

Keep the original claim, violating example, corrected reasoning/code and a fresh transfer result. If your first attempt is correct, critique a deliberate fault; do not invent personal errors to fill a quota. Distinguish proof-method preference from logical validity, and constructive examples from optimality arguments.