## Vacant sets and vacant nets: Component structures induced by a random walk

Given a discrete random walk on a finite graph G, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step t.%These sets induce subgraphs of the underlying graph. Let Γ(t) be the subgraph of Ginduced by the vacant set of the walk at step t. Similarly, let Γˆ(t) be the subgraph of G induced by the edges of the vacant net. For random r-regular graphs Gr, it was previously established that for a simple random walk, the graph Γ(t) of the vacant set undergoes a phase transition in the sense of the phase transition on Erd\H{os}-Renyi graphs Gn,p. Thus, for r≥3 there is an explicit value t∗=t∗(r) of the walk, such that for t≤(1−ϵ)t∗, Γ(t) has a unique giant component, plus components of size O(logn), whereas for t≥(1+ϵ)t∗ all the components of Γ(t) are of size O(logn). We establish the threshold value tˆ for a phase transition in the graph Γˆ(t) of the vacant net of a simple random walk on a random r-regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for r even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random r-regular graphs. The main findings are the following: As r increases the threshold for the vacant set converges to nlogr in all three walks. For the vacant net, the threshold converges to rn/2logn for both the simple random walk and non-backtracking random walk. When r≥4 is even, the threshold for the vacant net of the unvisited edge process converges to rn/2, which is also the vertex cover time of the process.