Hamiltonian properties of polyhedra with few 3-cuts : a survey

A

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

Every 4-connected graph with crossing number 2 is Hamiltonian

A seminal theorem of Tutte states that 4-connected planar graphs are Hamiltonian. Applying a result of Thomas and Yu, one can show that every 4-connected graph with crossing number 1 is Hamiltonian. In this paper, we continue along this path and prove the titular statement. We also discuss the traceability and Hamiltonicity of 3-connected graphs with small crossing number and few 3-cuts, and present applications of our results

Two Results in Drawing Graphs on Surfaces

In this work we present results on crossing-critical graphs drawn on non-planar surfaces and results on edge-hamiltonicity of graphs on the Klein bottle. We first give an infinite family of graphs that are 2-crossing-critical on the projective plane. Using this result, we construct 2-crossing-critical graphs for each non-orientable surface. Next, we use 2-amalgamations to construct 2-crossing-critical graphs for each orientable surface other than the sphere. Finally, we contribute to the pursuit of characterizing 4-connected graphs that embed on the Klein bottle and fail to be edge-hamiltonian. We show that known 4-connected counterexamples to edge-hamiltonicity on the Klein bottle are hamiltonian and their structure allows restoration of edge-hamiltonicity with only a small change

Hamiltonian Cycles in Polyhedral Maps

We present a necessary and sufficient condition for existence of a contractible, non-separating and noncontractible separating Hamiltonian cycle in the edge graph of polyhedral maps on surfaces. In particular, we show the existence of contractible Hamiltonian cycle in equivelar triangulated maps. We also present an algorithm to construct such cycles whenever it exists.Comment: 14 page