Graph Pebbling in Coq
This repository holds the formalization of various graph pebbling results, which I produced over the course of my master's thesis (1). The Coq source is split over multiple files which are ordered by their prefix. The prefix starts with a a letter that indicates to which part the file belongs. There are four parts:
- Part A - Utility library
- Part B - General graph pebbling
- Part C - Concrete pebbling bounds
- Part D - Number theory constructions
This formalization was developed using Coq 8.16.0 and Coq-stdpp 1.8.0. To check the formalization, install the stdpp library (2) and run these commands:
coq_makefile -f _CoqProject -o CoqMakefile
make -f CoqMakefile
Results
Pebbling bound of a diameter-2 graph
Hypothesis undirected : Symmetric E.
Hypothesis weight2 : ∀ u v, weight G u v = 2.
Hypothesis diameter2 : ∀ u v, u = v ∨ E u v ∨ ∃ w, E u w ∧ E w v.
Theorem pebbling_bound_diameter_2 :
pebbling_bound G (card V + 1).
Unidirectional solutions via pebble flows
Corollary pebble_flow_spec t n c :
(∃ c', c -->* c' ∧ n ≤ c' t) ↔
(∃ f, feasible c f ∧ unidirectional f ∧ n ≤ excess t c f).
Pebbling bound of an n-cube with generalized weights
Hypothesis ks_ge2 : Forall (λ k, k ≥ 2) ks.
Hypothesis ks_ord : ordered (≤) ks.
Corollary pebbling_bound_hypercube :
pebbling_bound (hypercube n ks) (product ks).
Pebbling bound of n in the divisor lattice of n
Corollary vertex_pebbling_bound_divisor_lattice n (non_zero : Premise (n > 0)) :
vertex_pebbling_bound (divisor_lattice n) n (top_divisor n).
The Erdős-Lemke conjecture
Corollary Erdos_Lemke_conjecture (l : list nat) d n :
length l = d -> d > 0 -> n > 0 -> d ∣ n -> (∀ a, a ∈ l -> a ∣ n) ->
∃ l', l' ≠ [] ∧ l' ⊆+ l ∧ d ∣ summation l' ∧ summation l' ≤ n.