On a Lower Bound for Short Noncontractible Cycles in Embedded Graphs

In this paper, a technique is developed that allows the construction of a triangulation of a closed orientable surface of genus $g$ by an $n$-vertex graph in such a way that the triangulation does not have short noncontractible cycles. Using this technique, a counterexample is constructed to a conjecture by Hutchinson that the length of the shortest noncontractible cycle in any such triangulation is $O(\sqrt{n/g} )$. The presented technique can also be used to show that the function $\sqrt{n/g} \log^{ *} g$ provides a lower bound for the shortest noncontractible cycle in a triangulation of a surface of genus $g$.