If W is the simple random walk on the square lattice Z2, then W induces a random walk WG on any spanning subgraph G ⊂ Z2 of the lattice as follows: viewing W as a uniformly random infinite word on the alphabet {x, −x, y, −y}, the walk WG starts at the origin and follows the directions specified by W, only accepting steps of W along which the walk WG does not exit G. For any fixed G ⊂ Z2, the walk WG is distributed as the simple random walk on G, and hence WG is almost surely recurrent in the sense that WG visits every site reachable from the origin in G infinitely often. This fact naturally leads us to ask the following: does W almost surely have the property that WG is recurrent for every G ⊂ Z2? We answer this question negatively, demonstrating that exceptional subgraphs exist almost surely. In fact, we show more to be true: exceptional subgraphs continue to exist almost surely for a countable collection of independent simple random walks, but on the other hand, there are almost surely no exceptional subgraphs for a branching random walk.@en

The deck of a graph (Formula presented.) is given by the multiset of (unlabeled) subgraphs (Formula presented.). The subgraphs (Formula presented.) are referred to as the cards of (Formula presented.). Brown and Fenner recently showed that, for (Formula presented.), the number of edges of a graph (Formula presented.) can be computed from any deck missing 2 cards. We show that, for sufficiently large (Formula presented.), the number of edges can be computed from any deck missing at most (Formula presented.) cards.@en

We construct asymptotically optimal adjacency labelling schemes for every hereditary class containing 2?(n2) n-vertex graphs as n? ?. This regime contains many classes of interest, for instance perfect graphs or comparability graphs, for which we obtain an adjacency labelling scheme with labels of n/4+o(n) bits per vertex. This implies the existence of a reachability labelling scheme for digraphs with labels of n/4+o(n) bits per vertex and comparability labelling scheme for posets with labels of n/4+o(n) bits per element. All these results are best possible, up to the lower order term. @en

Two permutations σ and π are ℓ-similar if they can be decomposed into subpermutations σ(1), . . . ,σ(ℓ) and π(1), . . . ,π(ℓ) such that σ(i) is order-isomorphic to π(i) for all i ∈[ℓ]. Recently, Dudek, Grytczuk, and Ruciński Variations on twins in permutations, Electron. J. Combin., 28 (2021), P3.19. posed the problem of determining the minimum ℓ for which two permutations chosen independently and uniformly at random are ℓ-similar. We show that two such permutations are O(n1/3 log11/6(n))-similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalizes to simultaneous decompositions of multiple permutations.@en

The deck of a graph G is the multiset of cards {G−v:v∈V(G)}. Myrvold (1992) showed that the degree sequence of a graph on n≥7 vertices can be reconstructed from any deck missing one card. We prove that the degree sequence of a graph with average degree d can be reconstructed from any deck missing O(n/d3) cards. In particular, in the case of graphs that can be embedded on a fixed surface (e.g. planar graphs), the degree sequence can be reconstructed even when a linear number of the cards are missing.@en