Local Algorithms, Regular Graphs of Large Girth, and Random Regular Graphs

Regular Graphs of given Girth

Graph theory is the study of mathematical structures called graphs. We define a graph as a pair (V,E), where V is a nonempty set, and E is a set of unordered pairs of elements of V . V is called the set of vertices of G, and E is the set of edges. Two vertices a and b are adjacent provided (a, b) ∈ E. If a pair of vertices is adjacent, the vertices are referred to as neighbors. We can represent...

Large independent sets in regular graphs of large girth

Let G be a d-regular graph with girth g, and let α be the independence number of G. We show that α(G) ≥ 12 ( 1− (d− 1)−2/(d−2) − (g) ) n where (g) → 0 as g → ∞, and we compute explicit bounds on (g) for small g. For large g this improves previous results for all d ≥ 7. The method is by analysis of a simple greedy algorithm which was motivated by the differential equation method used to bound in...

Independent sets and cuts in large-girth regular graphs

We present a local algorithm producing an independent set of expected size 0.44533n on large-girth 3-regular graphs and 0.40407n on large-girth 4-regular graphs. We also construct a cut (or bisection or bipartite subgraph) with 1.34105n edges on large-girth 3regular graphs. These decrease the gaps between the best known upper and lower bounds from 0.0178 to 0.01, from 0.0242 to 0.0123 and from ...

Small Regular Graphs of Girth 7

In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of (q + 1, 8)-cages, for q a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs. We obtain (q + 1)-regular graphs of girth 7...

On (minimal) regular graphs of girth 6