80 research outputs found
A sufficient condition guaranteeing large cycles in graphs
AbstractWe generalize Bedrossian-Chen-Schelp's condition (1993) for the existence of large cycles in graphs, and give infinitely many examples of graphs which fulfill the new condition for hamiltonicity, while the related condition by Bedrossian, Chen, and Schelp is not fulfilled
Oriented Colouring Graphs of Bounded Degree and Degeneracy
This paper considers upper bounds on the oriented chromatic number, ,
of graphs in terms of their maximum degree and/or their degeneracy
. In particular we show that asymptotically,
where and . This improves a result of MacGillivray, Raspaud, and
Swartz of the form . The rest of the paper is
devoted to improving prior bounds for in terms of and by
refining the asymptotic arguments involved.Comment: 8 pages, 3 figure
Pattern overlap implies runaway growth in hierarchical tile systems
We show that in the hierarchical tile assembly model, if there is a
producible assembly that overlaps a nontrivial translation of itself
consistently (i.e., the pattern of tile types in the overlap region is
identical in both translations), then arbitrarily large assemblies are
producible. The significance of this result is that tile systems intended to
controllably produce finite structures must avoid pattern repetition in their
producible assemblies that would lead to such overlap. This answers an open
question of Chen and Doty (SODA 2012), who showed that so-called
"partial-order" systems producing a unique finite assembly *and" avoiding such
overlaps must require time linear in the assembly diameter. An application of
our main result is that any system producing a unique finite assembly is
automatically guaranteed to avoid such overlaps, simplifying the hypothesis of
Chen and Doty's main theorem
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