---
tag: 28M8XD8EA9
kind: concept
name: Words and profiles
msc: [05-XX]
specializes: [0EPAT51DXQ]
sources:
  - arxiv: 2610.13556v1
    locator: Definition 2.1
    mention: ax://paper/2610.13556#TOY4
    licence: cc-by
    note: the paper that states the item (corpus arxiv AX9.2)
history:
  notes: The source is arXiv:2610.13556v1 (2026-10), by Jiang Jiang, Yuhan Jiang, Yiyang Li, Lujia Wang, Aitao Zhang. A paper also restates older results, so this is not the first statement until a person or a later pass confirms it.
inventory: [86VNXJWJ14]
uses: [0NCPXH194Y, 0VHJRXNAAS]
---

## Summary

A word on a graph $G$ is any map $w:V(G)\to\mathcal C$ into a set of colors, proper or not. Its profile is the vector $\mathbf m(w)$ whose entry $m_X(w)=|w^{-1}(X)|$ counts the vertices that get color $X$.

Take $|\mathcal C|=\chi(G)$, the [[chromatic number|0VHJRXNAAS]] of $G$. Then each proper [[colouring|0NCPXH194Y]] with palette $\mathcal C$ uses every color, and a profile is feasible if some such coloring has it.