Abstract
It is known that there are planar graphs without cycles of length 5 that are not 3-colorable. However, it was conjectured that every planar graph without cycles of length 5 is 3-colorable if it has no 14-cycles (Steinberg’s conjecture); or2intersecting triangles (the weak Bordeaux conjecture); or3adjacent triangles (the strong Bordeaux conjecture).All these conjectures remain open. As a variation of these conjectures, this paper proves that every planar graph without cycles of length 5 can be decomposed into a matching and a 3-colorable graph. This is the best possible in the sense that there are infinite planar graphs which have no such decomposition.
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