File size: 2,489 Bytes
e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import OAI.Combinatorics.HadwigerMatching.Definitions
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
namespace OAI
namespace HadwigerMatching
open scoped BigOperators
universe u
variable {V : Type u} {G : SimpleGraph V}
/-- The empty family is a touching matching. -/
theorem isTouchingMatching_empty (G : SimpleGraph V) :
IsTouchingMatching G ∅ := by
simp [IsTouchingMatching]
/-- Every member contributes exactly two vertices. -/
theorem IsTouchingMatching.sum_card_eq {M : Finset (Finset V)}
(hM : IsTouchingMatching G M) :
(∑ e ∈ M, e.card) = 2 * M.card := by
rw [Finset.sum_const_nat (fun e he => (hM.1 e he).1)]
exact Nat.mul_comm M.card 2
/-- Disjointness bounds twice the number of matching edges by the graph order. -/
theorem IsTouchingMatching.two_mul_card_le [Fintype V]
{M : Finset (Finset V)} (hM : IsTouchingMatching G M) :
2 * M.card ≤ Fintype.card V := by
classical
have hd : (M : Set (Finset V)).PairwiseDisjoint (fun e => e) := by
intro e he f hf hne
exact hM.2.1 e he f hf hne
have hc : (M.biUnion (fun e => e)).card = 2 * M.card := by
rw [Finset.card_biUnion hd]
exact hM.sum_card_eq
rw [← hc]
exact Finset.card_le_univ _
/-- Touching matching cardinalities are bounded on a finite vertex type. -/
theorem connectedMatchingNumber_bddAbove [Fintype V] (G : SimpleGraph V) :
BddAbove {n : ℕ | ∃ M : Finset (Finset V),
IsTouchingMatching G M ∧ M.card = n} := by
refine ⟨Fintype.card V, ?_⟩
rintro n ⟨M, hM, rfl⟩
have h : M.card + M.card ≤ Fintype.card V := by
simpa only [two_mul] using hM.two_mul_card_le
exact (Nat.le_add_right M.card M.card).trans h
/-- Every touching matching is bounded by the connected matching number. -/
theorem matching_card_le_number [Fintype V] {M : Finset (Finset V)}
(hM : IsTouchingMatching G M) : M.card ≤ connectedMatchingNumber G := by
exact le_csSup (connectedMatchingNumber_bddAbove G) ⟨M, hM, rfl⟩
/-- The connected matching number is realized by a touching matching. -/
theorem exists_matching_card_eq_number [Fintype V] (G : SimpleGraph V) :
∃ M : Finset (Finset V), IsTouchingMatching G M ∧
M.card = connectedMatchingNumber G := by
have hne : Set.Nonempty {n : ℕ | ∃ M : Finset (Finset V),
IsTouchingMatching G M ∧ M.card = n} :=
⟨0, ∅, isTouchingMatching_empty G, rfl⟩
exact Nat.sSup_mem hne (connectedMatchingNumber_bddAbove G)
end HadwigerMatching
end OAI
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