Investigations on graph-theoretical constructions in Homotopy type theory

{-# OPTIONS --without-K --exact-split #-}

module lib.graph-classes.PartiteGraph
  where
  open import foundations.Core

  open import lib.graph-definitions.Graph
  open Graph

  open import lib.graph-homomorphisms.Hom
  open import lib.graph-relations.Isomorphic

  module _ {ℓ : Level} (n : ℕ)
    where
    open import foundations.Finite
    open import lib.graph-families.CompleteGraph ℓ

    n-Partite :  Graph ℓ → Type (lsuc ℓ)
    n-Partite G = ∑[ A ∶ Graph ℓ ] (Hom G A × ∥ A ≅ (𝕂 n) ∥)

    Bipartite : (G : Graph ℓ) → Type (lsuc ℓ)
    Bipartite G = n-Partite G

-- conjecture1 : (G : Graph₀) → ∥ Node G ∥ → isSet (Bipartite₂ G)

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