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Lax17.Degree

Degree bounds

concepts/Lax17/Degree.lean · lax-17

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    Definition

    For a finite simple graph, a vertex has degree at most dd when its neighbourhood has at most dd vertices. A graph has maximum degree at most dd when this holds at every vertex.

    The definition is phrased through a finite set representing the neighbourhood. It therefore does not require choosing a decidable adjacency relation.

    Lean source view on GitHub

    1import Mathlib.Combinatorics.SimpleGraph.Finite
    2
    3/-!
    4---
    5title: Degree bounds
    6type: definition
    7---
    8For a finite simple graph, a vertex has degree at most \(d\) when its
    9neighbourhood has at most \(d\) vertices. A graph has maximum degree at most
    10\(d\) when this holds at every vertex.
    11
    12The definition is phrased through a finite set representing the neighbourhood.
    13It therefore does not require choosing a decidable adjacency relation.
    14-/
    15
    16namespace Lax17.Degree
    17
    18universe u
    19
    20/-- `N` is the neighbourhood of `v` in `G`. -/
    21def IsNeighbourhood {V : Type u} (G : SimpleGraph V) (v : V)
    22 (N : Finset V) : Prop :=
    23 ∀ w : V, w ∈ N ↔ G.Adj v w
    24
    25/-- The degree of `v` in `G` is at most `d`. -/
    26def AtMost {V : Type u} (G : SimpleGraph V) (v : V) (d : ℕ) : Prop :=
    27 ∃ N : Finset V, IsNeighbourhood G v N ∧ N.card ≤ d
    28
    29/-- The degree of `v` in `G` is exactly `d`. -/
    30def Exactly {V : Type u} (G : SimpleGraph V) (v : V) (d : ℕ) : Prop :=
    31 ∃ N : Finset V, IsNeighbourhood G v N ∧ N.card = d
    32
    33/-- Every vertex of `G` has degree at most `d`. -/
    34def MaximumAtMost {V : Type u} (G : SimpleGraph V) (d : ℕ) : Prop :=
    35 ∀ v : V, AtMost G v d
    36
    37end Lax17.Degree
    38

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