3-Coloring → Interval k-graph recognition without a partition Complexity open

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Clone https://github.com/reduction-zoo/unspecified-interval-partition to investigate 3-Coloring → Interval k-graph recognition without a partition. Read the fixed question and state. Use preparation notes, the test corpus, and verifier if present; otherwise complete Prepare before a reduction attempt. Follow the repository instructions and your AutoResearch pipeline.

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Why this question matters

This isolates the difference between recognizing a representation with supplied classes and discovering the classes.

Recognition with a supplied partition does not answer whether a compatible partition and interval representation can be found together.

The reduction to construct

The source gives a simple graph. Its outputs are proper vertex colorings with at most three colors, or NO-SOLUTION.

The target gives a graph and k. Find at most k vertex classes and one closed interval per vertex so that adjacency holds exactly for intersecting intervals in different classes, or report NO-SOLUTION. Endpoint weak-order ranks give a finite encoding.

Construct deterministic polynomial-time maps F and G: F sends every legal source instance to a legal target instance, and G(x,y) is a valid source output for every valid output y of F(x). Preserve the stated threshold, domain and promises. The requested complexity conclusion is NP-hardness for the stated target problem.

Both class assignments and interval order are free, so a supplied-partition reduction does not settle the target.

What a complete answer must include

Give explicit construction and recovery algorithms, a general proof for every legal input and every valid target output, and polynomial runtime and encoding-size bounds. Specify finite output encodings and handle NO-SOLUTION outputs when applicable. Tests compare recovered source outputs with independent source solutions.

References

Literature checked 2026-09-16. This summarizes the archived literature search on the date above. Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.