From 90f348ac1e77c9e655f5f3ba519bd3c210dc5653 Mon Sep 17 00:00:00 2001 From: jt496 <john.talbot@gmail.com> Date: Tue, 28 Jan 2025 16:37:30 +0000 Subject: [PATCH] colorable_zero_iff --- Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/Mathlib/Combinatorics/SimpleGraph/Coloring.lean b/Mathlib/Combinatorics/SimpleGraph/Coloring.lean index 54483becc0906..3a7e6f287d154 100644 --- a/Mathlib/Combinatorics/SimpleGraph/Coloring.lean +++ b/Mathlib/Combinatorics/SimpleGraph/Coloring.lean @@ -137,6 +137,10 @@ theorem isEmpty_of_colorable_zero (h : G.Colorable 0) : IsEmpty V := by obtain ⟨i, hi⟩ := h.some v exact Nat.not_lt_zero _ hi +@[simp] +lemma colorable_zero_iff : G.Colorable 0 ↔ IsEmpty V := + ⟨fun h => G.isEmpty_of_colorable_zero h, fun _ => G.colorable_of_isEmpty 0⟩ + /-- The "tautological" coloring of a graph, using the vertices of the graph as colors. -/ def selfColoring : G.Coloring V := Coloring.mk id fun {_ _} => G.ne_of_adj