Advances in International Applied Mathematics
Advances in International Applied Mathematics. 2026; 8: (2) ; 10.12208/j.aam.20260011 .
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南京航空航天大学数学学院 江苏南京
*通讯作者： 张旺凯,单位：南京航空航天大学数学学院 江苏南京; ；
Bickle在研究平面图三角剖分的退化度时，提出如下问题：给定整数𝑘,𝑙，是否存在退化度为𝑘的平面图𝐺，使得𝐺的任意三角剖分都有退化度𝑙，其中0≤𝑘≤5，3≤𝑙≤5。对于上述问题，除了(𝑘,𝑙)=(2,5)和(𝑘,𝑙)=(3,5)两种情形外，Bickle确定了其它所有情形。在本文中，我们肯定地回答(3,5)这种情形：证明存在一个3-退化平面图𝐺，使得𝐺的每一个三角剖分的退化度都是5。
When studying the degeneracy of triangulations of plane graphs, Bickle asked whether there exists a plane graph 𝐺 of degeneracy 𝑘 such that every triangulation of 𝐺 has degeneracy 𝑙, where 0≤𝑘≤5 and 3≤𝑙≤5. Except for the cases (𝑘,𝑙)=(2,5) and (𝑘,𝑙)=(3,5), the problem has been completely resolved. In this paper, we answer the case (3, 5) affirmatively. More precisely, we construct a 3-degenerate plane graph 𝐺 and prove that every triangulation of 𝐺 contains a 5-core. Consequently, every triangulation of 𝐺 has degeneracy 5.
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