On two conjectures on triangulations of 2-manifolds

Abstract

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-Poincar\'e 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 $\xi(T) = |V(T)| = V_{\min}(M)$, where $V_{\min}(M)$ denotes the minimum possible number of vertices among all triangulations of the 2-manifold $M$, and where $\xi(T)$ 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.