FormalConjecturesForMathlib.Combinatorics.SimpleGraph.Decomposition

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module


public import Mathlib.Combinatorics.SimpleGraph.Subgraph
public import Mathlib.Data.Set.Pairwise.Basic

@[expose] public section

Edge decompositions of simple graphs

SimpleGraph.IsDecomposition G D says that the finite family D of subgraphs of G partitions the edge set of G. Used by Erdős Problems 184 and 583.

namespace SimpleGraph

D is a decomposition of G into edge-disjoint subgraphs: the edge sets of the members of D are pairwise disjoint and their union is the edge set of G.

def IsDecomposition {V : Type*} (G : SimpleGraph V) (D : Finset G.Subgraph) : Prop :=
  Set.PairwiseDisjoint (D : Set G.Subgraph) (fun H ↦ H.edgeSet) ∧
  (⋃ H ∈ D, H.edgeSet) = G.edgeSet

end SimpleGraph