在一些充分条件下的不含4圈和7圈的平面图是2-弱退化的

Every Planar Graph with Some Sufficient Conditions but without 4- and 7-Cycles Is Weakly 2-Degenerate

摘要: 如果一个图G的每个子图中有一个点v，它的度数最多为k，那么我们称图G为k-退化的。k-弱退化图是k-退化图的推广。在这篇文章中，我们证明了不含4圈，7圈和特殊圈的平面图是2-弱退化的， 同时也是3-DP-可染的。由此得到结论：每个不含4圈，k圈，7圈和9圈的平面图是2-弱退化的以及3-DP-可染的，这里k ∈ {5, 6}。

Abstract: A graph G is k-degenerate if its every subgraph contains a vertex of degree at most k. Weakly k-degenerate graphs are a generalization of k-degenerate graphs. In this paper, we prove that every planar graph without 4, 7-cycles and some special cycles is weakly 2-degenerate. Consequently, it is 3-DP-colorable. As corollaries, every planar graph without 4-, k-, 7- and 9-cycles is weakly 2-degenerate and 3-DP-colorable, where k ∈ {5, 6}.