Abstract
Let G be a planar graph with neither 3-cycles nor adjacent 4-cycles. We prove that if G is connected and δ(G) ≥ 2, then G contains an edge uv with d(u) + d(v) ≤ 7 or a 2-alternating cycle. By this result, we obtain that G’s linear 2-arboricity \({la_{2}(G)\leq\lceil\frac{\Delta(G)+1}{2}\rceil+4.}\).
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This work is supported by a research grant Natural Science Foundation of Shandong Province (ZR2009AM009).
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Niu, HX., Cai, JS. Linear 2-Arboricity of Planar Graphs with Neither 3-Cycles Nor Adjacent 4-Cycles. Graphs and Combinatorics 29, 661–667 (2013). https://doi.org/10.1007/s00373-011-1122-2
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DOI: https://doi.org/10.1007/s00373-011-1122-2