On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

On 1-Hamilton-connected claw-free graphs

A graph G is k-Hamilton-connected (k-hamiltonian) if G−X is Hamilton-connected (hamiltonian) for every set X ⊂ V (G) with |X| = k. In the paper, we prove that (i) every 5-connected claw-free graph with minimum degree at least 6 is 1-Hamilton-connected, (ii) every 4-connected claw-free hourglass-free graph is 1-Hamilton-connected. As a byproduct, we also show that every 5-connected line graph with minimum degree at least 6 is 3-hamiltonian

Image Analyses of Two Crustacean Exoskeletons and Implications of the Exoskeletal Microstructure on the Mechanical Behavior

The microstructures of exoskeletons from Homarus americanus (American lobster) and Callinectes sapidus (Atlantic blue crab) were investigated to elucidate the mechanical behavior of such biological composites. Image analyses of the cross-sectioned exoskeletons showed that the two species each have three well-defined regions across the cuticle thickness where the two innermost regions (exocuticle and endocuticle) are load bearing. These regions consist of mineralized chitin fibers aligned in layers, where a gradual rotation of the fiber orientation of the layers results in repeating stacks. The exocuticle and endocuticle of the two species have similar morphology, but different thicknesses, number of layers, and number of stacks. Mechanics-based analyses showed that the morphology of the layered structure corresponds to a nearly isotropic structure. The cuticles are inter-stitched with pore canal fibers, running transversely to the layered structure. Mechanics-based analyses suggested that the pore canal fibers increase the interlaminar strength of the exoskeleton

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures