Is it true that if a graph is bipartite iff it is class 1 (edge-coloring)?

by mandella   Last Updated January 14, 2018 13:20 PM

A graph is class 1 if its edge set can be colored with $\Delta(G)$ colors, where $\Delta(G)$ is the maximum degree over all vertices in G. A graph is class 2 if we need $\Delta(G)+1$ colors to color the edge set. I think that if a graph is bipartite then we can color its edges with $\Delta(G)$ colors, while when it is not bipartite this cannot be done.

Is there any counterexample for this? Or is it true?

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