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Can someone tell me how to prove : The edge chromatic number of a graph must be at least ∆,where ∆ is the largest vertex degree of the graph???

Thank you for any advice😊

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Suppose (on the contrary) that the edge chromatic number is less than the largest vertex degree, say $d_{max}$. There exists some vertex, say $v$ with degree $d_{max}$ in the graph. Consider all edges of $v$ of $v$. If we colour all $d_{max}$ edges that have $v$ as an end, then we are bound to use at least $d_{max}$ colours, since if we use less than $d_{max}$ colours, at least 2 edges will have the same colour (by Pigeonhole principle). Hence edge chromatic number ( fewest number of colors necessary to color each edge of such that no two edges incident on the same vertex have the same color) is at least $d_{max}$

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  • $\begingroup$ This is a good answer, but as a matter of style, the proof by contradiction is unnecessary. (You still have a valid proof if you omit the first sentence.) $\endgroup$ – Misha Lavrov Nov 21 '18 at 17:26
  • $\begingroup$ @MishaLavrov thanks for the suggestion, I too think now that the first line is avoidable. Btw, what's so interesting about $e^{\sqrt{\log x}}$? $\endgroup$ – ab123 Nov 21 '18 at 22:04
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    $\begingroup$ It's just a weird intermediate function that grows slower than any polynomial that's always surprising when it occurs in asymptotic bounds. (Notably, Behrend's construction for 3AP-free sets.) $\endgroup$ – Misha Lavrov Nov 21 '18 at 22:15

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