Conjecture of TxGraffiti: Independence, domination, and matchings

TL;DR

This paper discusses an automated program, TxGraffiti, that generates and proves conjectures about graph invariants like independence, domination, and matching numbers, including their inequalities and sharpness.

Contribution

It proves and generalizes several conjectures generated by TxGraffiti on key graph invariants, expanding understanding of their relationships.

Findings

Proved and generalized conjectures on independence, domination, and matching numbers.

Established the sharpness of several proposed inequalities.

Enhanced the theoretical understanding of graph invariant relationships.

Abstract

TxGraffiti is an automated conjecturing program that produces graph theoretic conjectures in the form of conjectured inequalities. This program written and maintained by the second author since 2017 was inspired by the successes of previous automated conjecturing programs including Fajtlowicz's GRAFFITI and DeLaVi\~{n}a's GRAFFITI.pc. In this paper we prove and generalize several conjectures generated by TxGraffiti when it was prompted to conjecture on the \emph{independence number}, the \emph{domination number}, and the \emph{matching number} (and generalizations of each of these graph invariants). Moreover, in several instances we also show the proposed inequalities relating these graph invariants are sharp.