Lax Lean Archive

Multigraph representation of a simple graph

Lax683916.SimpleGraphMultigraphRepresentation · concepts/Lax683916/SimpleGraphMultigraphRepresentation.lean · lax-683916

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    Natural Language Statement

    Theorem

    Every simple graph has a simple loopless multigraph representation with the same vertices. The actual edges of the multigraph correspond bijectively to the unordered edges of the simple graph, and the correspondence preserves endpoints.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

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    1import Lax683916.MultigraphIsomorphism
    2
    3/-!
    4---
    5title: Multigraph representation of a simple graph
    6type: theorem
    7---
    8Every simple graph has a simple loopless multigraph representation with the
    9same vertices. The actual edges of the multigraph correspond bijectively to
    10the unordered edges of the simple graph, and the correspondence preserves
    11endpoints.
    12
    13# Formalization notes
    14
    15The statement gives one multigraph witness together with its simplicity and
    16the vertex-and-edge isomorphism. Its edge type is the canonical type `Sym2 V`
    17of all unordered vertex pairs; only graph edges belong to its actual edge
    18set. This is the converse of the multigraph-to-simple-graph representation
    19theorem.
    20-/
    21
    22namespace Lax683916.SimpleGraphMultigraphRepresentation
    23
    24open Lax683916.MultigraphIsomorphism
    25
    26universe u
    27
    28/-- Every simple graph is represented by a simple multigraph on its vertices. -/
    29axiom exists_multigraph_representation {V : Type u} (H : SimpleGraph V) :
    30 ∃ G : Graph V (Sym2 V), G.Simple ∧ Nonempty (IsomorphicToSimpleGraph G H)
    31
    32end Lax683916.SimpleGraphMultigraphRepresentation
    33
    Show Proof
    Formalization notes

    The statement gives one multigraph witness together with its simplicity and the vertex-and-edge isomorphism. Its edge type is the canonical type Sym2VSym2 V of all unordered vertex pairs; only graph edges belong to its actual edge set. This is the converse of the multigraph-to-simple-graph representation theorem.

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