A candidate proof of Chen's δ-complement total-domination Nordhaus–Gaddum conjecture (Conjecture 4)
Year: 2026
Type: preprint
Abstract: This note gives a candidate proof of Conjecture 4 of P. Chen, "Refined Nordhaus–Gaddum-TypeBounds for Roman and Total Domination on δ-Complement Graphs" (Mathematics 2026, 14(11):1797,doi:10.3390/math14111797). The conjecture states that for a finite simple graph G with kdistinct vertex degrees, if both G and its δ-complement G_δ are isolate-free, thenγ_t(G) + γ_t(G_δ) ≤ n + 2k. Chen proves it on a subclass and leaves the general case open.The proof here handles the general case via a degree-class trichotomy, a universal-vertexargument, the classical total-domination Nordhaus–Gaddum sum bound, and a charge/rescueaccounting; a construction of the two total dominating sets is given explicitly. IMPORTANT — status: This is a CANDIDATE proof. It was produced with the assistance of an AIsystem and re-checked by independent computer verification (source code included), but it hasNOT yet been reviewed by a human graph theorist. It should be read as a draft offered forrefereeing, not as an established result. Feedback, corrections, and counterexamples arewelcome. See §7 of the note for the honest caveats.
Author: SAFRI, Mohamed Rida
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7169666590