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Fedor Petrov
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Yes. For weak colorings, we may replace each edge with at least 2 vertices to an edge with exactly 2 vertices. It remains to properly color a graph with $n$ edges with $n$ colors. This is done by induction: color any vertex of any edge in a color $n-1$, remove the edges incident to this vertex and proceed.

Fedor Petrov
  • 108.9k
  • 9
  • 264
  • 459