14,465 research outputs found
Toughness and hamiltonicity in -trees
We consider toughness conditions that guarantee the existence of a hamiltonian cycle in -trees, a subclass of the class of chordal graphs. By a result of Chen et al.\ 18-tough chordal graphs are hamiltonian, and by a result of Bauer et al.\ there exist nontraceable chordal graphs with toughness arbitrarily close to . It is believed that the best possible value of the toughness guaranteeing hamiltonicity of chordal graphs is less than 18, but the proof of Chen et al.\ indicates that proving a better result could be very complicated. We show that every 1-tough 2-tree on at least three vertices is hamiltonian, a best possible result since 1-toughness is a necessary condition for hamiltonicity. We generalize the result to -trees for : Let be a -tree. If has toughness at least then is hamiltonian. Moreover, we present infinite classes of nonhamiltonian 1-tough -trees for each $k\ge 3
On realization graphs of degree sequences
Given the degree sequence of a graph, the realization graph of is the
graph having as its vertices the labeled realizations of , with two vertices
adjacent if one realization may be obtained from the other via an
edge-switching operation. We describe a connection between Cartesian products
in realization graphs and the canonical decomposition of degree sequences
described by R.I. Tyshkevich and others. As applications, we characterize the
degree sequences whose realization graphs are triangle-free graphs or
hypercubes.Comment: 10 pages, 5 figure
Stability of dynamical distribution networks with arbitrary flow constraints and unknown in/outflows
A basic model of a dynamical distribution network is considered, modeled as a
directed graph with storage variables corresponding to every vertex and flow
inputs corresponding to every edge, subject to unknown but constant inflows and
outflows. We analyze the dynamics of the system in closed-loop with a
distributed proportional-integral controller structure, where the flow inputs
are constrained to take value in closed intervals. Results from our previous
work are extended to general flow constraint intervals, and conditions for
asymptotic load balancing are derived that rely on the structure of the graph
and its flow constraints.Comment: published in proceeding of 52nd IEEE Conference on Decision and
Control (CDC 2013). arXiv admin note: text overlap with arXiv:1403.5198,
arXiv:1403.520
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, -tough graphs, and claw-free graphs
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