On the Girth of Graph Lifts

Abstract

The size of the smallest $k$-regular graph of girth at least $g$ is denoted by the well-studied function $n(k,g)$. We introduce an analogous function $n(H,g)$, defined as the smallest size graph of girth at least $g$ that is a lift (or cover) of the, possibly non-regular, graph $H$. We prove that the two main combinatorial bounds on $n(k,g)$ -- the Moore lower bound and the Erd\"{o}s-Sachs upper bound -- carry over to the new lift setting. We also consider two other functions: i) The smallest size graph of girth at least $g$ sharing a universal cover with $H$. We prove that it is the same as $n(H,g)$ up to a multiplicative constant. ii) The smallest size graph of girth least $g$ with a prescribed degree distribution. We discuss this known generalization and argue that the new suggested definitions are superior. We conclude with experimental results for a specific base graph, followed by conjectures and open problems for future research.