Edge count of a small complete graph
ProvedWilliamsonShmoys.small_card_complete_graph_edge_count__p2m_6ce4ca43_3f24_470a_9f5f_602e4f1a05ae_a1approximation-algorithmsplanar-graphstreewidth
For a finite linearly ordered vertex type with fewer than three elements, the complete graph's edge finset has cardinality equal to the binomial coefficient of the vertex count and 2.
Preamble
import Mathlib set_option autoImplicit false
Formal statement
theorem WilliamsonShmoys.small_card_complete_graph_edge_count__p2m_6ce4ca43_3f24_470a_9f5f_602e4f1a05ae_a1
{V : Type} [Fintype V] [LinearOrder V]
[DecidableRel (⊤ : SimpleGraph V).Adj]
(hsmall : Fintype.card V < 3) :
(⊤ : SimpleGraph V).edgeFinset.card = (Fintype.card V).choose 2 := by
sorrySource
David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).