Complete bipartite graphs

Content created by Egbert Rijke, Fredrik Bakke, Jonathan Prieto-Cubides, Vojtěch Štěpančík and Victor Blanchi.

Created on 2022-07-03.
Last modified on 2024-03-23.

module graph-theory.complete-bipartite-graphs where
Imports 
open import foundation.coproduct-types
open import foundation.universe-levels
open import foundation.unordered-pairs

open import graph-theory.finite-graphs

open import univalent-combinatorics.2-element-types
open import univalent-combinatorics.cartesian-product-types
open import univalent-combinatorics.coproduct-types
open import univalent-combinatorics.dependent-pair-types
open import univalent-combinatorics.fibers-of-maps
open import univalent-combinatorics.finite-types

Idea

Consider two finite sets X and Y. The complete bipartite graph on X and Y is the undirected finite graph consisting of:

  • The finite set of vertices is the coproduct type X + Y.

  • Given an unordered pair f : I → X + Y of vertices, the finite type of edges on the unordered pair (I , f) is given by

      (Σ (x : X), fiber f (inl x))  × (Σ (y : Y), fiber f (inr y)).

    In other words, an unordered pair of elements of the coproduct type X + Y is an edge in the complete bipartite graph on X and Y precisely when one of the elements of the unordered pair comes from X and the other comes from Y.

Definition

module _
  {l1 l2 : Level} (X : 𝔽 l1) (Y : 𝔽 l2)
  where

  vertex-finite-type-complete-bipartite-Undirected-Graph-𝔽 : 𝔽 (l1 ⊔ l2)
  vertex-finite-type-complete-bipartite-Undirected-Graph-𝔽 = coproduct-𝔽 X Y

  vertex-complete-bipartite-Undirected-Graph-𝔽 : UU (l1 ⊔ l2)
  vertex-complete-bipartite-Undirected-Graph-𝔽 =
    type-𝔽 vertex-finite-type-complete-bipartite-Undirected-Graph-𝔽

  unordered-pair-vertices-complete-bipartite-Undirected-Graph-𝔽 :
    UU (lsuc lzero ⊔ l1 ⊔ l2)
  unordered-pair-vertices-complete-bipartite-Undirected-Graph-𝔽 =
    unordered-pair vertex-complete-bipartite-Undirected-Graph-𝔽

  edge-finite-type-complete-bipartite-Undirected-Graph-𝔽 :
    unordered-pair-vertices-complete-bipartite-Undirected-Graph-𝔽 → 𝔽 (l1 ⊔ l2)
  edge-finite-type-complete-bipartite-Undirected-Graph-𝔽 p =
    product-𝔽
      ( Σ-𝔽 X
        ( λ x →
          fiber-𝔽
            ( finite-type-2-Element-Type (pr1 p))
            ( coproduct-𝔽 X Y)
            ( element-unordered-pair p)
            ( inl x)))
      ( Σ-𝔽 Y
        ( λ y →
          fiber-𝔽
            ( finite-type-2-Element-Type (pr1 p))
            ( coproduct-𝔽 X Y)
            ( element-unordered-pair p)
            ( inr y)))

  edge-complete-bipartite-Undirected-Graph :
    unordered-pair-vertices-complete-bipartite-Undirected-Graph-𝔽 → UU (l1 ⊔ l2)
  edge-complete-bipartite-Undirected-Graph p =
    type-𝔽 (edge-finite-type-complete-bipartite-Undirected-Graph-𝔽 p)

  complete-bipartite-Undirected-Graph-𝔽 :
    Undirected-Graph-𝔽 (l1 ⊔ l2) (l1 ⊔ l2)
  pr1 complete-bipartite-Undirected-Graph-𝔽 =
    vertex-finite-type-complete-bipartite-Undirected-Graph-𝔽
  pr2 complete-bipartite-Undirected-Graph-𝔽 =
    edge-finite-type-complete-bipartite-Undirected-Graph-𝔽

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