On the Halin Tur\'an number of short cycles

A Halin graph is a graph constructed by embedding a tree with no vertex of degree two in the plane and then adding a cycle to join the tree's leaves. The Halin Tur\'an number of a graph $F$, denoted as $\ex_{\hh}(n,F)$, is the maximum number of edges in an $n$-vertex Halin graph. In this paper, we give the exact value of $\ex_{\mathcal{H}}(n,C_4)$, where $C_4$ is a cycle of length 4. We also pose a conjecture for the Halin Tur\'an number of longer cycles.