Decomposing a planar graph without cycles of length 5 into a matching and a 3-colorable graph

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 1 es (Steinberg’s conjecture); or ecting triangles (the weak Bordeaux conjecture); or nt triangles (the strong Bordeaux conjecture). ese 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.