Given a graph (Formula presented.), a decomposition of (Formula presented.) is a partition of its edges. A graph is (Formula presented.) -decomposable if its edge set can be partitioned into a (Formula presented.) -degenerate graph and a graph with maximum degree at most (Formula presented.). For (Formula presented.), we are interested in the minimum integer (Formula presented.) such that every planar graph is (Formula presented.) -decomposable. It was known that (Formula presented.), (Formula presented.), and (Formula presented.). This paper proves that (Formula presented.), and (Formula presented.).
This study begun during the 5th Korean Early Career Researcher Workshop in Combinatorics. Eun\u2010Kyung Cho was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (No. NRF\u20102020R1I1A1A0105858711). Ilkyoo Choi was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (No. NRF\u20102018R1D1A1B07043049), and also by the Hankuk University of Foreign Studies Research Fund. Ringi Kim was supported by the National Research Foundation of Korea grant funded by the Korea government (No. NRF\u20102021R1C1C1010763), and also by INHA UNIVERSITY Research Grant. Boram Park was supported by the National Research Foundation of Korea grant funded by the Korea government (No. NRF\u20102018R1C1B6003577). Xuding Zhu was supported by NSFC 11971438, U20A2068, and ZJNSFC LD19A010001.
