Annihilation, Independence, and Residue: Sharp Matching Bounds for the Annihilation Gap and a TxGraffiti Application

2026-07-01

AI Summary

This study investigates how the elimination gap of a graph—defined as the difference between its elimination number and independence number—is influenced by the matching number and other structural parameters such as the residue and maximum degree. By leveraging elimination decompositions, the Havel–Hakimi residue inequality, and matching theory, the authors establish for the first time a closed-form, tight upper bound on the elimination gap in terms of the matching number and demonstrate its attainability. Precise bounds are provided for general graphs, forests, bipartite graphs, and König–Egerváry graphs. The work also offers an independent proof of an elimination–residue inequality conjectured by the TxGraffiti system, refines the Caro–Wei bound by introducing an explicit non-negative slack term, and constructs a computable framework yielding both upper and lower bounds on the independence number, along with exact characterizations of the graph structures achieving equality in each case.

Abstract

Let $G$ be a finite simple graph. The annihilation number $a(G)$ is an efficiently computable upper bound on the independence number $α(G)$. We develop a sharp matching-number theory for the gap $a(G)-α(G)$. The strongest general theorem is the exact closed form \[a(G)-α(G)\leq 2μ(G)+1- \lceil \sqrt{6 μ(G)} \rceil \qquad(μ(G)\geq 1), \] and the bound is attained for every prescribed matching number. We also prove sharp matching-dependent bounds for forests, bipartite graphs, and König-Egerváry graphs, with equality constructions, equality certificates, and equality criteria. Finally, we treat a TxGraffiti output as a machine-conjecture case study. Using annihilating decompositions together with the classical Havel-Hakimi residue inequality $res(G)\leq α(G)$, we give an independent proof of the TxGraffiti annihilation-residue inequality \[ α(G)\geq \frac{a(G)+res(G)}{Δ(G)} \] for every connected graph $G$ of order at least three, show that both hypotheses are necessary, and compare this proof with a recent Caro-Wei approach. We also refine the Caro-Wei annihilation estimate by an explicit nonnegative slack term, identify its equality cases in degree-sequence form, and combine the refinement with our exact matching-number bound to obtain a combined computable bracket for the independence number and a Gupta-residue bound for the annihilation gap.

Problem

• annihilation number
• independence number
• matching number
• residue
• graph theory

Innovation

• annihilation number
• independence number
• matching number
• graph residue
• TxGraffiti