Lower bound on independence via annihilation number and residue

Establish that for every nontrivial connected finite simple undirected graph G with independence number α(G), annihilation number a(G), residue R(G), and maximum degree Δ(G), the inequality α(G) ≥ (a(G) + R(G)) / Δ(G) holds; moreover, demonstrate that this bound is sharp.

Background

This conjecture connects three classical graph invariants: the independence number α(G), the annihilation number a(G), and the residue R(G). Prior work established α(G) ≤ a(G) and α(G) ≥ R(G). The conjecture proposes a new lower bound on α(G) that combines a(G) and R(G) and scales by the maximum degree Δ(G).

It emerged from early TxGraffiti sessions and has withstood attempts at proof or disproof. The authors note it holds for several structured families (e.g., regular bipartite graphs and cubic König–Egerváry graphs), underscoring its plausibility and sharpness.