124 research outputs found
Longest Paths in Circular Arc Graphs
As observed by Rautenbach and Sereni (arXiv:1302.5503) there is a gap in the
proof of the theorem of Balister et al. (Longest paths in circular arc graphs,
Combin. Probab. Comput., 13, No. 3, 311-317 (2004)), which states that the
intersection of all longest paths in a connected circular arc graph is
nonempty. In this paper we close this gap.Comment: 7 page
Critical percolation on random regular graphs
We show that for all the size of the largest
component of a random -regular graph on vertices around the percolation
threshold is , with high probability. This extends
known results for fixed and for , confirming a prediction of
Nachmias and Peres on a question of Benjamini. As a corollary, for the largest
component of the percolated random -regular graph, we also determine the
diameter and the mixing time of the lazy random walk. In contrast to previous
approaches, our proof is based on a simple application of the switching method.Comment: 10 page
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