Vizing's theorem states that a graph can be edge-colored in either Δ or Δ + 1 colors, where Δ is the maximum vertex degree of the graph. A graph with edge chromatic number equal to Δ is known as a class 1 graph. König's line coloring theorem states that all bipartite graphs are class 1. All cubic Hamiltonian graphs are class 1, as are planar graphs with maximum vertex degree Δ>7. Class 1 graphs have both edge chromatic number and fractional edge chromatic number equal to Δ.
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