By Xu B.G.

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**Extra info for A 3-color Theorem on Plane Graphs without 5-circuits**

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Sd+1 ], implicitly assuming that we have a ﬁxed linear order on the 0-cells in ∆. Whenever σ and τ are disjoint faces such that σ ∪ τ ∈ ∆, we deﬁne [σ] ∧ [τ ] in the natural manner. Note that [∅] ∧ z = z for all z. Boundary Map The boundary map ∂d : C˜d (∆; F) → C˜d−1 (∆; F) is the homomorphism deﬁned by d+1 ∂d ([s1 ] ∧ . . ∧ [sd+1 ]) = (−1)i−1 [s1 ] ∧ . . ∧ [si−1 ] ∧ [si+1 ] ∧ . . ∧ [sd+1 ]. 1). Combining all ˜ F) of all C˜d (∆; F). It is ∂d , we obtain an operator ∂ on the direct sum C(∆; 2 ˜ F), ∂) well-known and easy to see that ∂ = 0.

By some abuse of notation, we deﬁne the topological realization of ∆ as any topological space homeomorphic to the following space ∆ : Let e1 , . . , en be an orthonormal basis for Euclidean space Rn . For a face σ, let σ denote the set λx ex : x∈σ λx = 1, λx > 0 for all x ∈ σ . 5) x∈σ Deﬁne ∆ as the union σ∈∆ σ ; this is a disjoint union. Note that 2σ = τ ⊆σ τ ; this is the convex hull of the set {ex : x ∈ σ}. Also note that {x} = {ex }. We refer to ∆ as the canonical realization of ∆ Let ∆ and Γ be deﬁned on two disjoint vertex sets X and Y .

Trivial. Semipure shellable =⇒ Semipure constructible. The theorem is obvious if ∆ is a simplex. 26. Write ∆1 = fdel∆ (σ) and ∆2 = 2σ ∗ lk∆ (σ); it is clear that ∆ = ∆1 ∪ ∆2 . Now, ∆1 is semipure shellable by assumption. 30, ∆2 = 2σ ∗ lk∆ (σ) is semipure shellable. Finally, the intersection ∆1 ∩ ∆2 equals ∂2σ ∗ lk∆ (σ). The boundary of a simplex is well-known to be shellable and hence semipure shellable; hence ∆1 ∩ ∆2 is semipure shellable. Induction yields that all these complexes are semipure constructible.

### A 3-color Theorem on Plane Graphs without 5-circuits by Xu B.G.

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