Results on sphere maps, triangulations, states and Hamiltonian cycles, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Deleting an edge — Lemma 1.4. Deleting an edge merges the two faces of its darts or splits the face containing both, other faces unchanged.
• Sphere maps and bridges — Proposition 1.6. Every map has chi <= 2; in a sphere map an edge is a bridge iff its darts lie in one face, and deleting edges leaves sphere maps.
• Trees and cotrees — Lemma 2.2. The dual edges of the complement of a spanning tree of a sphere map form a spanning tree of the dual; for a spanning forest they form a connected spanning subgraph.
• Face boundaries span the cycle space — Lemma 3.1. In a sphere map the face boundaries span the cycle space over F2, with the single relation that all of them sum to zero.
• Two sides of a cycle — Theorem 3.2. A cycle of a sphere map splits the faces into two sides, an edge lying in the cycle iff its darts lie on different sides.
• Euler count for a side — Proposition 3.5. For a side of a cycle, (vertices strictly inside) - (edges inside) + (faces) = 1.
• Regions of a connected submap — Proposition 4.1. The faces of a connected submap correspond to the classes of faces joined across edges outside it.
• Dual triangulation — Theorem 2.3. The dual of a simple cubic bipartite 3-connected sphere map is a face-coloured sphere triangulation.
• Signed count — Lemma 3.2. On the side A of a cycle of length h in a face-coloured triangle map, 3(B - W) = 2l - h.
• Cutting along a separating triangle — Lemma 4.2. Cutting a sphere triangulation along a separating triangle gives two smaller coloured sphere triangulations.
• Two disjoint perfect matchings — Lemma 2.2. A bipartite graph satisfying the degree-two Hall condition has two edge-disjoint perfect matchings.
• Density inequality — Proposition 3.1. Without separating triangles, e(U) - b(U) <= 2|U| - 4 for every vertex set U containing the roots.
• Pairs of states exist (OpenAI) — Proposition 3.3. A coloured sphere triangulation without separating triangles has a pair of states.
• Disk identity (OpenAI) — Lemma 1.1. A cycle traversed along the chosen darts of a state has delta-sum -3 counterclockwise and 3 clockwise.
• Cancellation (OpenAI) — Lemma 2.2. For a state s with a cycle in Q_s, the signed sum of i^J(r,s) over its partners r vanishes.
• Face potentials — Lemma 3.1. A sphere map carries antisymmetric dart weights with face sums 1 off a chosen face; cycles with that face on the right get positive weight.
• A forest state (OpenAI) — Proposition 4.2. If a pair of states exists, some pair (r, s) has Q_s a forest.
• Two induced trees without separating triangles (OpenAI) — Theorem 4.1. A coloured sphere triangulation without separating triangles splits into two induced trees with a prescribed split of one face.
• Two induced trees (OpenAI) — Theorem 5.1. Every coloured sphere triangulation splits into two induced trees with a prescribed split of one face.
• Barnette's conjecture (OpenAI) — Theorem 6.1. Every simple cubic bipartite 3-connected graph carrying a sphere map has a Hamiltonian cycle avoiding any prescribed edge.

Reading order

Read the lessons in order; each uses the notation and results of the previous ones.