44 research outputs found
Hexagonal Tilings: Tutte Uniqueness
We develop the necessary machinery in order to prove that hexagonal tilings
are uniquely determined by their Tutte polynomial, showing as an example how to
apply this technique to the toroidal hexagonal tiling.Comment: 12 figure
Maximum and minimum toughness of graphs of small genus
AbstractA new lower bound on the toughness t(G) of a graph G in terms of its connectivity ϰ(G) and genus γ(G) is obtained. For γ > 0, the bound is sharp via an infinite class of extremal graphs all of girth 4. For planar graphs, the bound is t(G) > ϰ(G)/2 − 1. For ϰ = 1 this bound is not sharp, but for each ϰ = 3, 4, 5 and any ϵ > 0, infinite families of graphs {G(ϰ, ϵ)} are provided with ϰ(G(ϰ, ϵ)) = ϰ, but t(G(ϰ, ϵ)) < ϰ/2 − 1 + ϵ.Analogous investigations on the torus are carried out, and finally the question of upper bounds is discussed. Several unanswered questions are posed
Hexagonal Tilings and Locally C6 Graphs
We give a complete classification of hexagonal tilings and locally C6 graphs,
by showing that each of them has a natural embedding in the torus or in the
Klein bottle. We also show that locally grid graphs are minors of hexagonal
tilings (and by duality of locally C6 graphs) by contraction of a perfect
matching and deletion of the resulting parallel edges, in a form suitable for
the study of their Tutte uniqueness.Comment: 14 figure