Flanking Mumbers and Arankings of Cyclic Graphs

Given a graph \( G \), a \( k \)-ranking is a labeling of the vertices such that any path connecting two vertices with the same label contains a vertex with a larger label. A \( k \)-ranking is minimal if and only if reducing any label violates the ranking property. The arank number of a graph \( \psi_r(G) \), is the maximum \( k \) such that \( G \) has a minimal \( k \)-ranking. The arank number of a cycle was first investigated by Kostyuk and Narayan. They determined precise arank numbers for most cycles, and determined the arank number within \( 1 \) for all other cases. In this paper we introduce a new concept called the flanking number, which is used to solve all open cases. We prove that \( \psi_r(C_n) = \lfloor\log_2(n + 1)\rfloor + \lfloor\log_2 \left(\frac{n+2}{3}\right)\rfloor + 1 \) for all \( n > 6 \), which completely solves the problem that has been open since \( 2003 \).