Generation of cubic graphs

We describe a new algorithm for the efficient generation of all non-isomorphic connected cubic graphs. Our implementation of this algorithm is more than 4 times faster than previous generators. The generation can also be efficiently restricted to cubic graphs with girth at least 4 or 5

The domination parameters of cubic graphs

Let ir(G), γ(G), i(G), β0(G), Γ(G) and IR(G) be the irredundance number, the domination number, the independent domination number, the independence number, the upper domination number and the upper irredundance number of a graph G, respectively. In this paper we show that for any nonnegative integers k 1, k 2, k 3, k 4, k 5 there exists a cubic graph G satisfying the following conditions: γ(G) - ir(G) ≤ k 1, i(G) - γ(G) ≤ k 2, β0(G) - i(G) > k 3, Γ(G) - β0(G) - k 4, and IR(G) - Γ(G) - k 5. This result settles a problem posed in [9]. © Springer-Verlag 2005

Generation of cubic graphs and snarks with large girth

We describe two new algorithms for the generation of all non-isomorphic cubic graphs with girth at least $k\ge 5$ which are very efficient for $5\le k \le 7$ and show how these algorithms can be efficiently restricted to generate snarks with girth at least $k$. Our implementation of these algorithms is more than 30, respectively 40 times faster than the previously fastest generator for cubic graphs with girth at least 6 and 7, respectively. Using these generators we have also generated all non-isomorphic snarks with girth at least 6 up to 38 vertices and show that there are no snarks with girth at least 7 up to 42 vertices. We present and analyse the new list of snarks with girth 6.Comment: 27 pages (including appendix

On the Signed 22-independence Number of Graphs

In this paper, we study the signed 2-independence number in graphs and give new sharp upper and lower bounds on the signed 2-independence number of a graph by a simple uniform approach. In this way, we can improve and generalize some known results in this area

N=2 Gauge Theories: Congruence Subgroups, Coset Graphs and Modular Surfaces

We establish a correspondence between generalized quiver gauge theories in four dimensions and congruence subgroups of the modular group, hinging upon the trivalent graphs which arise in both. The gauge theories and the graphs are enumerated and their numbers are compared. The correspondence is particularly striking for genus zero torsion-free congruence subgroups as exemplified by those which arise in Moonshine. We analyze in detail the case of index 24, where modular elliptic K3 surfaces emerge: here, the elliptic j-invariants can be recast as dessins d'enfant which dictate the Seiberg-Witten curves.Comment: 42+1 pages, 5 figures; various helpful comments incorporate

Generation of Cubic Graphs

We describe a new algorithm for the efficient generation of all non-isomorphic connected cubic graphs. Our implementation of this algorithm is more than 4 times faster than previous generators. The generation can also be efficiently restricted to cubic graphs with girth at least 4 or 5

Dessins d'enfants in N=2 generalised quiver theories

We study Grothendieck’s dessins d’enfants in the context of the N=2 supersymmetric gauge theories in (3 + 1) dimensions with product SU (2) gauge groups which have recently been considered by Gaiotto et al.. We identify the precise context in which dessins arise in these theories: they are the so-called ribbon graphs of such theories at certain isolated points in the moduli space. With this point in mind, we highlight connections to other work on trivalent dessins, gauge theories, and the modular group