A Systematic Study of Single-Anchor Logical Gadgets

TL;DR

This paper systematically analyzes logical gadgets for 3-coloring with a single anchor, introducing a framework for ladgets, identifying minimal gadgets, and revealing the rarity of certain logical structures.

Contribution

It introduces a new framework for single-anchor logical gadgets, classifies minimal gadgets, and provides exhaustive analysis and embedding techniques for 3-coloring problems.

Findings

Identified exactly two minimal XNOR ladgets among 29 billion configurations.

Developed an embedding technique from 3-coloring ladgets to k-coloring.

Showed the single anchor constraint creates a fundamentally different framework.

Abstract

We present a systematic study of logical gadgets for 3-coloring under a single anchor constraint, where only one color representing logical falsehood is fixed to a vertex. We introduce a framework of what we call ladgets (logical gadgets), graph gadgets that implement Boolean functions. Then, we define a set of core gadgets, called primitives, which help identify and analyze the logical behavior of ladgets. Next, we examine the structure of several standard ladgets and present several structural constraints for ladgets. Through an exhaustive search of all non-isomorphic connected graphs up to 10 vertices, we verify all minimal constructions for standard ladgets. Notably, we identify exactly two non-isomorphic minimal XNOR ladgets in approximately 29 billion gadget configurations, highlighting the rarity of gadgets capable of expressing logical behavior. We also present an embedding technique that embeds ladgets with less than 3 inputs from 3-coloring into k-coloring. Our work shows how the single anchor constraint creates a fundamentally different framework from the two anchor gadgets used in SAT reductions.