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import FormalConjecturesUtilVizing's conjecture (1968)
References:
[Vi68] Vizing, V. G. (1968). "Some unsolved problems in graph theory." Uspekhi Mat. Nauk 23, pp. 117--134.
[ClSu00] Clark, W. E. and Suen, S. (2000). "An inequality related to Vizing's conjecture." Electron. J. Combin. 7, N4.
[SuTa12] Suen, S. and Tarr, J. (2012). "An improved inequality related to Vizing's conjecture." Electron. J. Combin. 19, P8.
[BDGHHKR12] Brešar, B., Dorbec, P., Goddard, W., Hartnell, B. L., Henning, M. A., Klavžar, S. and Rall, D. F. (2012). "Vizing's conjecture: a survey and recent results." J. Graph Theory 69, pp. 46--76.
open SimpleGraphnamespace VizingConjectureVizing's conjecture (1968).
For all finite simple graphs $G$ and $H$, the domination number of the Cartesian (box) product satisfies $\gamma(G ,\square, H) \ge \gamma(G),\gamma(H)$.
@[category research open, AMS 5]
theorem vizing_conjecture : answer(sorry) ↔
∀ {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β]
(G : SimpleGraph α) (H : SimpleGraph β),
G.dominationNumber * H.dominationNumber ≤ (G □ H).dominationNumber := ⊢ True ↔
∀ {α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β] (G : SimpleGraph α) (H : SimpleGraph β),
G.dominationNumber * H.dominationNumber ≤ (G □ H).dominationNumber
All goals completed! 🐙The Clark–Suen inequality (2000).
For all finite simple graphs $G$ and $H$, $\gamma(G ,\square, H) \ge \tfrac12,\gamma(G),\gamma(H)$; that is, Vizing's conjecture holds up to a factor of $2$.
Reference: [ClSu00].
@[category research solved, AMS 5]
theorem vizing_conjecture.variants.clark_suen
{α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β]
(G : SimpleGraph α) (H : SimpleGraph β) :
G.dominationNumber * H.dominationNumber ≤ 2 * (G □ H).dominationNumber := α:Typeβ:Typeinst✝³:Fintype αinst✝²:Fintype βinst✝¹:DecidableEq αinst✝:DecidableEq βG:SimpleGraph αH:SimpleGraph β⊢ G.dominationNumber * H.dominationNumber ≤ 2 * (G □ H).dominationNumber
All goals completed! 🐙The Suen–Tarr inequality (2012).
For all finite simple graphs $G$ and $H$, $\gamma(G ,\square, H) \ge \tfrac12,\gamma(G),\gamma(H) + \tfrac12\min{\gamma(G), \gamma(H)}$, improving the Clark–Suen bound.
Reference: [SuTa12].
@[category research solved, AMS 5]
theorem vizing_conjecture.variants.suen_tarr
{α β : Type} [Fintype α] [Fintype β] [DecidableEq α] [DecidableEq β]
(G : SimpleGraph α) (H : SimpleGraph β) :
G.dominationNumber * H.dominationNumber + min G.dominationNumber H.dominationNumber ≤
2 * (G □ H).dominationNumber := α:Typeβ:Typeinst✝³:Fintype αinst✝²:Fintype βinst✝¹:DecidableEq αinst✝:DecidableEq βG:SimpleGraph αH:SimpleGraph β⊢ G.dominationNumber * H.dominationNumber + min G.dominationNumber H.dominationNumber ≤ 2 * (G □ H).dominationNumber
All goals completed! 🐙
Vizing's conjecture when γ(H) = 1.
If H has a dominating vertex, so that $\gamma(H) = 1$, then Vizing's inequality reduces to
$\gamma(G) \le \gamma(G ,\square, H)$, which is the projection bound
dominationNumber_le_dominationNumber_boxProd. This is the simplest of the known cases of the
conjecture (see [BDGHHKR12]).
α:Typeβ:Typeinst✝³:Fintype αinst✝²:Fintype βinst✝¹:DecidableEq αinst✝:DecidableEq βG:SimpleGraph αH:SimpleGraph βhH:H.dominationNumber = 1hne:Nonempty β⊢ G.dominationNumber ≤ (G □ H).dominationNumber
All goals completed! 🐙end VizingConjecture