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

TL;DR

This paper establishes a high-probability lower bound on the size of maximum edge-cuts in large-girth and random cubic graphs, advancing understanding of their structural properties.

Contribution

It introduces a probabilistic method to estimate large edge-cuts in cubic graphs with large girth and in random cubic graphs, providing new bounds and insights.

Findings

Existence of a probability distribution on edge-cuts with each edge in the cut with probability ≥ 0.88672

Maximum edge-cut size at least 1.33008n in large-girth cubic graphs

Random cubic graphs contain an edge cut of size at least 1.33008n asymptotically almost surely

Abstract

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.