A graph G is k-degenerate if each subgraph of G has a vertex of degree at most k. It is known that every simple planar graph with girth 6, or equivalently without 3-, 4-, and 5-cycles, is 2-degenerate. In this work, we investigate for which k every planar graph without 4-, 6-, … , and 2k-cycles is 2-degenerate. We determine that k is 5 and the result is tight since the truncated dodecahedral graph is a 3-regular planar graph without 4-, 6-, and 8-cycles. As a related result, we also show that every planar graph without 4-, 6-, 9-, and 10-cycles is 2-degenerate.

中文翻译：

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没有特定循环的平面图是 2-退化的

一个图ģ是K-简并如果每个子图ģ具有至多度的顶点ķ。众所周知，每个周长为 6 或等效地没有 3、4 和 5 圈的简单平面图都是 2 退化的。在这项工作中，我们探讨这ķ每个平面图没有4,6，...  ，和2 ķ -cycles是2退化。我们确定k为 5 并且结果是紧的，因为截断的十二面体图是一个没有 4、6 和 8 圈的 3-正则平面图。作为相关结果，我们还表明，每个没有 4-、6-、9-和 10-循环的平面图都是 2-退化的。