On Compact Symmetric Regularizations of Graphs

Let G be a finite simple graph of order n, maximum degree ∆, and minimum degree δ. A compact regularization of G is a ∆-regular graph H of which G is an induced subgraph: H is symmetric if every automorphism of G can be extended to an automorphism of H. The index |H : G| of a regularization H of G is the ratio |V (H)|/|V (G)|. Let mcr(G) denote the index of a minimum compact regularization of G and let mcsr(G) denote the index of a minimum compact symmetric regularization of G. Erdős and Kelly proved that every graph G has a compact regularization and mcr(G) 6 2. Building on a result of König, Chartrand and Lesniak showed that every graph has a compact symmetric regularization and mcsr(G) 6 2∆−δ. Using a partial Cartesian product construction, we improve this to mcsr(G) 6 ∆− δ + 2 and give examples to show this bound cannot be reduced below ∆− δ + 1.