isBipartite — Mathlib

English

If G.IsBipartiteWith s t, then G is bipartite.

Русский

Если GIsBipartiteWith s t, то G двудольный.

LaTeX

$$G.IsBipartiteWith s t ⇒ G.IsBipartite$$

Lean4

/-- If a simple graph `G` has a bipartition, then it is bipartite. -/
theorem isBipartite {s t : Set V} (h : G.IsBipartiteWith s t) : G.IsBipartite :=
  by
  refine ⟨s.indicator 1, fun {v w} hw ↦ ?_⟩
  obtain (⟨hs, ht⟩ | ⟨ht, hs⟩) := h.2 hw <;>
    { replace ht : _ ∉ s := h.1.subset_compl_left ht; simp [hs, ht]
    }

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