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I found the best edge coloring for the next two graphs.

But is there some strategy for it?

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  • $\begingroup$ Edge coloring is NP-Hard. The greedy algorithm is usually a good place to start. I am not familiar with any approximation algorithms off the top of my head, but that is also another good avenue to pursue. $\endgroup$ – ml0105 Dec 25 '17 at 20:54
  • $\begingroup$ @ml0105 What do you mean with edge coloring is an NP-hard problem? Is Vertex coloring also NP-hard? $\endgroup$ – WinstonCherf Dec 26 '17 at 14:50
  • $\begingroup$ Yes. Unless P=NP, I wouldn’t expect an easy solution to exist in general. $\endgroup$ – ml0105 Dec 26 '17 at 15:49
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The main observation is that the edge-chromatic number will be at least the maximum degree, i.e., $\chi'(G) \geq \Delta(G)$, simply because each edge incident with a maximum-degree vertex must have different colors. In fact, Vizing's Theorem goes further and says $$\chi'(G) \in \{\Delta(G),\Delta(G)+1\}.$$

The first graph achieves $\chi'(G)=\Delta(G)$ (meaning it's class one), as you demonstrate. There's theoretical results which imply this, e.g., the Wikipedia page for Vizing's Theorem states "Vizing (1965) showed that a planar graph is of class one if its maximum degree is at least eight."

The second graph (the Petersen graph) is not class one.

There's not going to be an efficient algorithm for this in general, since there's various NP-completeness results, e.g. Leven and Galil, NP completeness of finding the chromatic index of regular graphs, 1983.

  • In the given examples, the most difficult task is proving $\chi'(G) \neq \Delta(G)$ in the Petersen graph, since it requires effort to check it can't be $3$-edge-colored.

  • A greedy-like algorithm would likely work in practice to give a $\chi'(G)$-edge coloring; generally, when there's one coloring, there's zillions of others.

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