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In 2003, O. V. Borodin and A. Raspaud conjectured that every planar graph with the minimum distance between triangles, dΔ, at least 1 and without 5‐cycles is 3‐colorable. This Bordeaux conjecture (Brdx3CC) has common features with Havel's (1970) and Steinberg's (1976) 3‐color problems. A relaxation of Brdx3CC with dΔ⩾4 was confirmed in 2003 by Borodin and Raspaud, whose result was improved to dΔ⩾3...
He, Hou, Lih, Shao, Wang, and Zhu showed that a planar graph of girth 11 can be decomposed into a forest and a matching. Borodin, Kostochka, Sheikh, and Yu improved the bound on girth to 9. We give sufficient conditions for a planar graph with 3-cycles to be decomposable into a forest and a matching.
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