11,387 research outputs found

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

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    Given a discrete random walk on a finite graph GG, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step tt.%These sets induce subgraphs of the underlying graph. Let Γ(t)\Gamma(t) be the subgraph of GG induced by the vacant set of the walk at step tt. Similarly, let Γ^(t)\widehat \Gamma(t) be the subgraph of GG induced by the edges of the vacant net. For random rr-regular graphs GrG_r, it was previously established that for a simple random walk, the graph Γ(t)\Gamma(t) of the vacant set undergoes a phase transition in the sense of the phase transition on Erd\H{os}-Renyi graphs Gn,pG_{n,p}. Thus, for r3r \ge 3 there is an explicit value t=t(r)t^*=t^*(r) of the walk, such that for t(1ϵ)tt\leq (1-\epsilon)t^*, Γ(t)\Gamma(t) has a unique giant component, plus components of size O(logn)O(\log n), whereas for t(1+ϵ)tt\geq (1+\epsilon)t^* all the components of Γ(t)\Gamma(t) are of size O(logn)O(\log n). We establish the threshold value t^\widehat t for a phase transition in the graph Γ^(t)\widehat \Gamma(t) of the vacant net of a simple random walk on a random rr-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 rr even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random rr-regular graphs. The main findings are the following: As rr increases the threshold for the vacant set converges to nlogrn \log r in all three walks. For the vacant net, the threshold converges to rn/2  lognrn/2 \; \log n for both the simple random walk and non-backtracking random walk. When r4r\ge 4 is even, the threshold for the vacant net of the unvisited edge process converges to rn/2rn/2, which is also the vertex cover time of the process.Comment: Added results pertaining to modified walk

    On giant components and treewidth in the layers model

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    Given an undirected nn-vertex graph G(V,E)G(V,E) and an integer kk, let Tk(G)T_k(G) denote the random vertex induced subgraph of GG generated by ordering VV according to a random permutation π\pi and including in Tk(G)T_k(G) those vertices with at most k1k-1 of their neighbors preceding them in this order. The distribution of subgraphs sampled in this manner is called the \emph{layers model with parameter} kk. The layers model has found applications in studying \ell-degenerate subgraphs, the design of algorithms for the maximum independent set problem, and in bootstrap percolation. In the current work we expand the study of structural properties of the layers model. We prove that there are 33-regular graphs GG for which with high probability T3(G)T_3(G) has a connected component of size Ω(n)\Omega(n). Moreover, this connected component has treewidth Ω(n)\Omega(n). This lower bound on the treewidth extends to many other random graph models. In contrast, T2(G)T_2(G) is known to be a forest (hence of treewidth~1), and we establish that if GG is of bounded degree then with high probability the largest connected component in T2(G)T_2(G) is of size O(logn)O(\log n). We also consider the infinite two-dimensional grid, for which we prove that the first four layers contain a unique infinite connected component with probability 11

    A sharp threshold for random graphs with a monochromatic triangle in every edge coloring

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    Let R\R be the set of all finite graphs GG with the Ramsey property that every coloring of the edges of GG by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let G(n,p)G(n,p) be the random graph on nn vertices with edge probability pp. We prove that there exists a function c^=c^(n)\hat c=\hat c(n) with 000 0, as nn tends to infinity Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is of independent interest is a generalization of Szemer\'edi's Regularity Lemma to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.
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