arXiv subjects

Pawel Pralat

Publications and source records attributed to Pawel Pralat.

74 records · Page 5Linked to original sources

arXiv2007

On the threshold for k-regular subgraphs of random graphs

The $k$-core of a graph is the largest subgraph of minimum degree at least $k$. We show that for $k$ sufficiently large, the $(k + 2)$-core of a random graph $\G(n,p)$ asymptotically almost surely has a spanning $k$-regular subgraph. Thus the threshold for the appearance of a $k$-regular subgraph of a random graph is at most the threshold for the $(k+2)$-core. In particular, this pins down the point of appearance of a $k$-regular subgraph in $\G(n,p)$ to a window for $p$ of width roughly $2/n$ for large $n$ and moderately large $k$.

math.CO↗

arXiv2006

Geodesics and almost geodesic cycles in random regular graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d_G(u,v) is at least d_C(u,v)-e(n). Let f(n) be any function tending to infinity with n. We consider a random d-regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n)= \log_{d-1} \log_{d-1} n +f(n) and |C|=2\log_{d-1}n+O(f(n)). Along the way, we obtain results on near-geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph.

math.MG↗