关于二维流形三角剖分的两个猜想
On two conjectures on triangulations of 2-manifolds
AI总结：
本文针对二维流形三角剖分的两个Chen-Lawrencenko猜想，利用曲面三角剖分的欧拉-庞加莱公式相关贪心算法证明了第一个猜想，并通过克莱因瓶的8顶点三角剖分反例反驳了第二个猜想。
AI中文摘要：
对于闭连通二维流形$M$及其三角剖分$T$，记$V(T)$为$T$的顶点集。三角剖分$T$的循环着色指的是$T$的面着色，满足：对每个$v \in V(T)$，与$v$关联的面具有不同颜色。Chen和Lawrencenko[《横滨数学杂志》，1999]提出，存在仅依赖于$M$的常数$C(M)$，使得$|V(T)| + C(M)$种颜色足以对$M$的三角剖分$T$进行循环着色。我们利用与曲面三角剖分的欧拉-庞加莱公式相关的贪心算法证明了该猜想。Chen和Lawrencenko还提出另一个猜想：若$M$不是射影平面，且$T$是$M$的顶点数最少的三角剖分，则$\xi(T) = |V(T)| = V_{\min}(M)$，其中$V_{\min}(M)$表示二维流形$M$所有三角剖分中顶点数的最小值，$\xi(T)$表示$T$可使用$k$种颜色进行循环着色的最小$k$值。我们通过一个明确的反例反驳了后一猜想，该反例为克莱因瓶的8顶点、16面三角剖分。在我们的工作之前，Chen-Lawrencenko的这两个猜想似乎仍未解决。
英文摘要：
For a closed, connected 2-manifold $M$, and for a triangulation $T$ of $M$, we write $V(T)$ in place of the vertex set associated with $T$. A cyclic coloration of a triangulation $T$ of $M$ refers to a face coloring of $T$ such that: For each $v \in V(T)$, the faces incident to $v$ have distinct colors. Chen and Lawrencenko [Yokohama Math. J., 1999] conjectured that there exists a constant $C(M)$ (depending only on $M$) such that $|V(T)| + C(M)$ colors suffice for there to exist a cyclic coloration of a triangulation $T$ of $M$. We prove this conjecture, using a greedy algorithm related to the Euler-Poincaré formula for surface triangulations. Chen and Lawrencenko also conjectured that: If $M$ is not the projective plane and $T$ is a triangulation of $M$ that is minimal with respect to the number of vertices, then $ξ(T) = |V(T)| = V_{\min}(M)$, where $V_{\min}(M)$ denotes the minimum possible number of vertices among all triangulations of the 2-manifold $M$, and where $ξ(T)$ denotes the minimal value $k$ such that $T$ admits a cyclic coloration with $k$ colors. We disprove this latter conjecture via an explicit counterexample, using an 8-vertex triangulation of the Klein bottle with 16 faces. It appears that both of the Chen-Lawrencenko conjectures have remained open, prior to our work.
