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  "body": "\n## Definition\n\nLet $G$ be a finite [[graph|0JD5C82QMB]] with [[chromatic number|0VHJRXNAAS]] $\\chi(G)$. Let $\\mathcal C$ be a set of colors with $|\\mathcal C|=\\chi(G)$.\n\n**Words and profiles** are defined as follows.\n\nA word on $G$ is any map $w:V(G)\\to\\mathcal C$. The word need not be a proper [[colouring|0NCPXH194Y]]: adjacent vertices may get the same color.\n\nThe profile of a word $w$ is the vector $\\mathbf m(w)=(m_X(w))_{X\\in\\mathcal C}$. Its entry $m_X(w)=|w^{-1}(X)|$ is the number of vertices of color $X$.\n\nVectors are written in bold, and inequalities between vectors hold in every coordinate. The vector $\\mathbf e_X$ has entry $1$ at $X$ and $0$ elsewhere. The vector $\\mathbf 1$ has every entry equal to $1$.\n\nLet $\\mathcal P(G)$ be the set of proper colorings of $G$ with palette $\\mathcal C$. Since $|\\mathcal C|=\\chi(G)$, each coloring in $\\mathcal P(G)$ uses every color.\n\nA profile $\\mathbf m$ is feasible if $\\mathbf m=\\mathbf m(w)$ for some $w\\in\\mathcal P(G)$.\n",
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      "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."
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        "locator": "Definition 2.1",
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        "note": "the paper that states the item (corpus arxiv AX9.2)"
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