63 research outputs found
Fault-tolerant Designs in Lattice Networks on the Klein Bottle
In this note, we consider triangular, square and hexagonal lattices on the flat Klein bottle, and find subgraphs with the property that for any vertices there exists a longest path (cycle) avoiding all of them. This completes work previously done in other lattices
Remarks Upon Recognising Genus and Possible Shapes of Chemical Cages in the Form of Polyhedra, Tori and Klein Bottles
Some of the problems associated with recognising and classifying cage
structures are reviewed briefly and discussed. Some new structures are
considered, including Klein bottles (polyhex and azulenoidl and \u27neal\u27 polyhex double tori
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
Towards obtaining a 3-Decomposition from a perfect Matching
A decomposition of a graph is a set of subgraphs whose edges partition those
of . The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011
states that every connected cubic graph can be decomposed into a spanning tree,
a 2-regular subgraph, and a matching. It has been settled for special classes
of graphs, one of the first results being for Hamiltonian graphs. In the past
two years several new results have been obtained, adding the classes of plane,
claw-free, and 3-connected tree-width 3 graphs to the list.
In this paper, we regard a natural extension of Hamiltonian graphs: removing
a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely,
removing a perfect matching from a cubic graph leaves a disjoint union
of cycles. Contracting these cycles yields a new graph . The graph is
star-like if is a star for some perfect matching , making Hamiltonian
graphs star-like. We extend the technique used to prove that Hamiltonian graphs
satisfy the 3-decomposition conjecture to show that 3-connected star-like
graphs satisfy it as well.Comment: 21 pages, 7 figure
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