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  "body": "\n## Definition\n\nLet $G$ be a finite [[graph|0JD5C82QMB]].\nFor each natural number $k$, let $P(G, k)$ be the number of [[colourings|0874138YP6]] of $G$ with the set of colours $\\{1, \\dots, k\\}$.\nThere is a unique polynomial $P(G, x)$ whose value at each natural number $k$ is $P(G, k)$. It is the chromatic polynomial of $G$.\n\n## Remarks\n\nFor the [[path graph|075WR6W3Y3]] $P_3$, $P(P_3, x) = x(x-1)^2$. For the [[complete graph|03WKB2H474]], $P(K_n, x) = x(x-1)\\cdots(x-n+1)$.\nFor an edge $e$, the deletion–contraction rule says $P(G, x) = P(G - e, x) - P(G / e, x)$. Here $G / e$ merges the two ends of $e$.\nThe [[chromatic number|0C5M7AM6FH]] is the smallest positive integer $k$ with $P(G, k) \u003e 0$.\n\n## Explanations\n\n### L1 human\n\nThe chromatic polynomial counts the proper colourings of a graph with $k$ colours, for each $k$.\nGeorge David Birkhoff introduced it in 1912 to study the four colour problem.\n",
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