Discharging cartwheels

In [J. Combin. Theory Ser. B 70 (1997), 2-44] we gave a simplified proof of the Four Color Theorem. The proof is computer-assisted in the sense that for two lemmas in the article we did not give proofs, and instead asserted that we have verified those statements using a computer. Here we give additional details for one of those lemmas, and we include the original computer programs and data as “ancillary files” accompanying this submission. 30 April 1995. Revised 4 February 1997 ∗ Research partially performed under a consulting agreement with Bellcore, and partially supported by DIMACS Center, Rutgers University, New Brunswick, New Jersey 08903, USA. 1 Partially supported by NSF under Grant No. DMS-8903132 and by ONR under Grant No. N0001492-J-1965. 2 Partially supported by DIMACS and by ONR under Grant No. N00014-93-1-0325. 3 Partially supported by NSF under Grant No. DMS-9303761 and by ONR under Grant No. N0001493-1-0325. 1. AXLES We assume familiarity with [1]. The purpose of this manuscript is to give a description of the program that we used to establish [1, Theorem (7.1)]. We begin by showing that it suffices to prove the equivalent of [1, Theorem (7.1)] for parts (K, a, b), where a, b : V (K) → {5, 6, . . . , 12}. While this is not really necessary, it makes the computer programs slightly simpler and more elegant. We say that a part (K, a, b) is limited if a(v) ≤ 12 for every v ∈ V (K). A trivial limited part is a part (K, a, b) such that b(v) = 5 and a(v) = 12 for every vertex v of K except the hub. It is unique up to isomorphism. (1.1) Let d = 7, 8, 9, 10, 11. If the trivial limited part with hub of degree d is successful, then so is the trivial part with hub of degree d. Proof. Let W be a cartwheel with hub of degree d. Let G be obtained from G(W ) by replacing every fan over a vertex of valency > 12 by an 8-edge path, and let W ′ be the cartwheel with G(W ) = G and γW ′(v) = min{γ(v), 12} if v ∈ V (G(W )) ∩ V (G ) and γW ′(v) = 12 if v ∈ V (G ) − V (G(W )). Then the trivial limited part of degree d fits W , and hence either a good configuration appears in W , or NP(W ) ≤ 0. Since every good configuration K satisfies γK(v) ≤ 11 for every v ∈ V (K), and NP(W ) = NP(W ) by condition (iv) in the definition of a rule and the fact that δ(v) ∈ {5, 6, 7, 8,∞} for every rule (G, β, δ, r, s, t) in [1, Figure 5] and every v ∈ V (G), we deduce that either a good configuration appears in W , or NP(W ) ≤ 0, as required. If (K, a, b) is a part, then K is, up to isomorphism, determined by the mappings a, b. We now make this precise. Let d be an integer. An axle of degree d is a pair A = (l, u), where l, u : {1, 2, . . . , 5d} → {5, 6, . . . , 12} such that (A1) l(i) ≤ u(i) for every i = 1, 2 . . . , 5d, 2 (A2) l(i) ∈ {5, 6, 7, 8, 9} and u(i) ∈ {5, 6, 7, 8, 12} for all i = 1, 2, . . . , 5d, and (A3) for i = 1, 2, . . . , d, if l(i) 6= u(i), then (l(j), u(j)) = (5, 12) for j = 2d+ i, 3d+ i, 4d+ i. We write lA = l and uA = u, and put lA(0) = uA(0) = d. Let A be an axle of degree d, and let (K, a, b) be a part such that (P1) the hub of K is 0, (P2) the spokes of K are 1, 2, . . . , d in order, (P3) the hats of K are d+ 1, d+ 2, . . . , 2d in order (so that for i = 1, 2, . . . , d− 1, d+ i is adjacent to i and i+ 1, and 2d is adjacent to 1 and d), (P4) for i = 1, 2, . . . , d, if k = l(i) = u(i), then (5 ≤ k ≤ 8 by (A2) and) the vertices of the fan over i are 2d+ i, 3d+ i, . . . , (k − 4)d+ i in order (so that 2d+ i, 3d+ i, . . . , (k − 4)d+ i, d+ i form a path in K in order; if k = 5 there are no fan vertices), and (P5) b, a are the restrictions of l, u to V (K), respectively. In these circumstances we say that (K, a, b) is the part derived from A. It is unique up to isomorphism. A condition is a pair (n,m), where n ∈ {1, 2, . . . , 5d} and m ∈ {−8,−7,−6,−5, 6, 7, 8, 9}. We say that a condition (n,m) is compatible with an axle A if (C1) lA(n) ≤ −m < uA(n) if m < 0, (C2) lA(n) < m ≤ uA(n) if m > 0, and (C3) either n ≤ 2d, or n = jd+ i, where j ∈ {2, 3, 4}, i ∈ {1, 2, . . . , d} and lA(i) = uA(i) ≥ j + 4. If (n,m) is a condition we define ¬(n,m) to be the condition (n, 1−m). It follows immediately that (n,m) is compatible with an axle if and only if ¬(n,m) is. Let A be an axle, and let c = (n,m) be a condition compatible with A. We define 3 (l, u) by l(i) = { lA(i) if i 6= n or m < 0 m otherwise u(i) = { uA(i) if i 6= n or m > 0 −m otherwise. It follows that (l, u) is an axle; we put A ∧ c = (l, u). It follows immediately that if A is an axle and c is a condition compatible with A, then the parts derived from A ∧ c and A∧ (¬c) are a complementary pair of refinements of the part derived from A. We say that an axle is successful if the part derived from it is successful. By (1.1) we can restate [1, Theorem (7.1)] as follows. An axle A of degree d is trivial if (lA(i), uA(i)) = (5, 12) for all i = 1, 2, . . . , 5d. We denote the trivial axle of degree d by Ωd. (1.2) For d = 7, 8, 9, 10, 11 the trivial axle of degree d is successful. Let W be a cartwheel and A an axle, both of degree d. We say that W is compatible with A if the part derived from A fits W . It is easy to see that W is compatible with A if and only if W satisfies (P1), (P2), (P3), (P4), and lA(n) ≤ γW (n) ≤ uA(n) for all n ∈ {0, 1, . . . , 2d} and all n of the form n = jd+ i, where i ∈ {1, 2, . . . , d}, lA(i) = uA(i) and j = 2, 3, . . . , lA(i) − 4. We say that an axle A is reducible if for every cartwheel W compatible with A a good configuration appears in W .