A Systematic Study of Single-Anchor Logical Gadgets

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 structural constraints applicable to all 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 graphs capable of expressing logical behavior. We also present a general embedding technique that embeds ladgets 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.