Short Even Cycles in Cages With Odd Girth

A \((k;g)\)-cage is a smallest \(k\)-regular graph with girth \(g\). Harary and Kovacs [2] conjectured that for all \(k \geq 3\) and odd \(g \geq 5\), there exists a \((k;g)\)-cage which contains a cycle of length \(g+1\). Among other results, we prove the conjecture for all \(k \geq 3\) and \(g \in \{5,7\}\).