Graphs with Independent Exact $r$-covers for all $r$
Hou Tin Chau
TL;DR
The paper resolves Gray and Johnson's question by constructing finite $d$-regular graphs that contain independent exact $r$-covers for all $r\in[1,d]$, using a two-step approach: build base graphs with a single independent exact $r$-cover and combine them via finite common coverings. It develops a robust framework of generalized graphs, covering maps, and a diagonal tensor-product construction to produce common coverings with controlled vertex counts, leveraging 1-/2-factor decompositions (Petersen's theorem) and Schreier-coset perspectives to optimize structure. The authors derive divisibility conditions on the graph order $n$ and determine exact minimal orders for $d=3,4,5,6$ (namely $40,210,6048,9240$ vertices) via explicit constructions from small base graphs and common coverings. They also provide asymptotic growth bounds for the minimal order $N(d)$ and discuss open cases (notably $d=7$) and potential extensions beyond pairwise cover intersections.
Abstract
For every natural number $d$, we construct finite $d$-regular simple graphs that, for every $r \le d$, contain an independent exact $r$-cover. This answers a question of Gray and Johnson that arose in their study of 2-step transit probabilities. We obtain some divisibility conditions on the order $n$ of graphs that for every $r \le d$ contain an independent exact $r$-cover, and give constructions for $d=3, 4, 5, 6$ where the order of the graph is minimal (we deduce this minimality from our divisibility conditions). We construct these graphs as common coverings of smaller graphs. We revisit a result of Angluin and Gardiner on finite common coverings of two regular graphs of the same degree, and the result of Gross that regular graphs of even degree are Schreier coset graphs. We combine both results to provide a finite common covering of two regular graphs of the same degree, that uses fewer vertices than the construction of Angluin and Gardiner in some cases.