An annihilation-number Caro-Wei bound: a TxGraffiti conjecture and an independence-number bracket

Abstract

Automated conjecturing programs scan collections of graphs for inequalities between invariants that no stored graph violates, then offer the survivors for proof or refutation. TxGraffiti, one such program, conjectured that every nontrivial connected graph $G$ satisfies $\alpha(G) \ge \bigl(a(G) + R(G)\bigr)/\Delta(G)$, where $\alpha$ is the independence number, $a$ the annihilation number, $R$ the residue, and $\Delta$ the maximum degree. Established only for two special families of graphs, the conjecture has otherwise remained open. The note proves the degree-sequence inequality $a \le \tfrac{\Delta+1}{2}W$, where $W$ is the Caro-Wei sum; the same inequality is known for the independence number in place of $a$. Combined with the classical lower bounds $\alpha \ge R$ and $\alpha \ge W$, it proves the conjecture for every connected graph of maximum degree at least three, and a direct argument settles maximum degree two; the conjecture fails only for the single edge, of maximum degree one. The inequality also brackets the independence number between the polynomial-time quantities $R$ and $a$, within a factor $(\Delta+1)/2$. The conjecture's bound is sharp, with equality attained, for instance, by the complete graph on four vertices.