An Upper Bound for the \((n,5)\) – cages

A \((n,5)\)-cage is a minimal graph of regular degree \(n\) and girth \(5\). Let \(f(n,5)\) denote the number of vertices in a \((n,5)\)-cage. The best known example of an \((n,5)\)-cage is the Petersen graph, the \((3,5)\)-cage. The \((4,5)\)-cage is the Robertson graph, the \((7,5)\)-cage is the Hoffman-Singleton graph, the \((6,5)\)-cage was found by O’Keefe and Wong~[2] and there are three known \((5,5)\)-cages. No other \((n,5)\)-cages are known for \(n \geq 8\). In this paper, we will use a graph structure called remote edges and a set of mutually orthogonal Latin squares to give an upper bound of \(f(n,5)\) for \(n = 2^k+1\).