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  "body": "\n## Statement\n\nLet $G$ be a finite simple [[graph|0JD5C82QMB]] with [[maximum degree|0KNAN9F27P]] $\\Delta$.\nLet $\\chi'(G)$ be the chromatic index of $G$, the least number of colours in a proper [[edge colouring|02CDYWSWSY]] of $G$.\nThen\n$$\\Delta \\le \\chi'(G) \\le \\Delta + 1.$$\n\n## Remarks\n\nThe lower bound is clear: the $\\Delta$ edges at a vertex of maximum degree need distinct colours.\nThe upper bound is proved by colouring the edges one at a time with $\\Delta + 1$ colours.\nWhen a new edge cannot be coloured, a fan of edges at one end and a path in two colours are recoloured to free a colour.\nGraphs with $\\chi' = \\Delta$ are of class 1, and graphs with $\\chi' = \\Delta + 1$ are of class 2.\nAn odd cycle has $\\Delta = 2$ and needs $3$ colours, so it is of class 2; every bipartite graph is of class 1, by a theorem of Kőnig.\nThe hypothesis that $G$ is simple is needed: the multigraph on three vertices with each pair joined by $k$ parallel edges has $\\Delta = 2k$ but needs $3k$ colours.\nDeciding whether a graph is of class 1 is NP-complete.\n\n## Explanations\n\n### L1 human\n\nColour the edges of a graph so that edges at the same vertex get different colours.\nIf some vertex has $\\Delta$ edges, you need at least $\\Delta$ colours.\nVizing's theorem says that one extra colour is always enough.\n\n### L2 human\n\nFor vertex colouring, the chromatic number can be far above any simple lower bound, but for edge colouring Vizing's theorem pins the answer to two values.\nThe proof is constructive and gives an efficient algorithm for a colouring with $\\Delta + 1$ colours.\nFor multigraphs, Vizing also proved $\\chi' \\le \\Delta + \\mu$, where $\\mu$ is the largest number of parallel edges.\n",
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