Maximum edge-cuts in cubic graphs with large girth and in random cubic graphs

We show that for every cubic graph G with sufficiently large girth there exists a probability distribution on edge-cuts of G such that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that G contains an edge-cut of size at least 1.33008n, where n is the number of vertices of G, and has fractional cut covering number at most 1.127752. The lower bound on the size of maximum edge-cut also applies to random cubic graphs. Specifically, a random n-vertex cubic graph a.a.s. contains an edge cut of size 1.33008n

On the minimum bisection of random 3−3-regular graphs

In this paper we give new asymptotically almost sure lower and upper bounds on the bisection width of random $3-$regular graphs. The main contribution is a new lower bound on the bisection width of $0.103295n$, based on a first moment method together with a structural decomposition of the graph, thereby improving a 27 year old result of Kostochka and Melnikov. We also give a complementary upper bound of $0.139822n$, combining known spectral ideas with original combinatorial insights. Developping further this approach, with the help of Monte Carlo simulations, we obtain a non-rigorous upper bound of $0.131366n$.Comment: 48 pages, 20 figure