Skip to content

feat(GraphTheory/ExampleSheet1): question 3 #54

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
YaelDillies merged 2 commits into from
Sep 2, 2026
Merged

feat(GraphTheory/ExampleSheet1): question 3 #54

Changes from all commits
Commits
Show all changes
2 commits
Select commit Hold shift + click to select a range
da4dbf3
feat(GraphTheory/ExampleSheet1): prove question 3
n33levo Aug 11, 2026
b5ad1a1
Merge upstream master into nsarkar/prove-equal-vertex-degrees
n33levo Sep 2, 2026
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
21 changes: 18 additions & 3 deletions CamCombi/GraphTheory/ExampleSheet1.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -56,9 +56,24 @@ description of all graphs that satisfy this equality?
Show that every graph $$G$$, with $$|G| > 2$$, has two vertices of the same degree.
-/

lemma q3 [Fintype α] (G : SimpleGraph α) [DecidableRel G.Adj] :
∃ a b, a ≠ b ∧ G.degree a = G.degree b :=
sorry
lemma q3 [Fintype α] [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj] :
∃ a b, a ≠ b ∧ G.degree a = G.degree b := by
suffices ¬Injective (fun v ↦ G.degree v) by
obtain ⟨a, b, hab, hne⟩ :=
(Function.not_injective_iff (f := fun v ↦ G.degree v)).mp this
exact ⟨a, b, hne, hab⟩
intro hdegree
let degreeFin : α → Fin (card α) := fun v ↦ ⟨G.degree v, G.degree_lt_card_verts v⟩
have hdegreeFin : Injective degreeFin := fun _ _ h ↦ hdegree <| congrArg Fin.val h
have hdegreeFin_bijective : Bijective degreeFin :=
(Fintype.bijective_iff_injective_and_card degreeFin).2 ⟨hdegreeFin, by simp⟩
obtain ⟨a, ha⟩ := hdegreeFin_bijective.surjective ⟨0, Fintype.card_pos⟩
obtain ⟨b, hb⟩ := hdegreeFin_bijective.surjective
⟨card α - 1, Nat.sub_lt Fintype.card_pos (by simp)⟩
have ha_isolated : G.IsIsolated a := (G.degree_eq_zero a).mp <| congrArg Fin.val ha
have hb_universal : G.IsUniversal b :=
(G.degree_eq_card_sub_one b).mp <| congrArg Fin.val hb
exact hb_universal.not_isIsolated a ha_isolated

/-!
### Question 4
Expand Down
Loading