7,549 research outputs found
The degree-diameter problem for sparse graph classes
The degree-diameter problem asks for the maximum number of vertices in a
graph with maximum degree and diameter . For fixed , the answer
is . We consider the degree-diameter problem for particular
classes of sparse graphs, and establish the following results. For graphs of
bounded average degree the answer is , and for graphs of
bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases
for fixed . For graphs of given treewidth, we determine the the maximum
number of vertices up to a constant factor. More precise bounds are given for
graphs of given treewidth, graphs embeddable on a given surface, and
apex-minor-free graphs
Lower bound theorems for general polytopes
For a -dimensional polytope with vertices, , we
calculate precisely the minimum possible number of -dimensional faces, when
or . This confirms a conjecture of Gr\"unbaum, for these
values of . For , we solve the same problem when or ; the
solution was already known for . In all these cases, we give a
characterisation of the minimising polytopes. We also show that there are many
gaps in the possible number of -faces: for example, there is no polytope
with 80 edges in dimension 10, and a polytope with 407 edges can have dimension
at most 23.Comment: 26 pages, 3 figure
On the maximum order of graphs embedded in surfaces
The maximum number of vertices in a graph of maximum degree and
fixed diameter is upper bounded by . If we
restrict our graphs to certain classes, better upper bounds are known. For
instance, for the class of trees there is an upper bound of
for a fixed . The main result of
this paper is that graphs embedded in surfaces of bounded Euler genus
behave like trees, in the sense that, for large , such graphs have
orders bounded from above by begin{cases} c(g+1)(\Delta-1)^{\lfloor
k/2\rfloor} & \text{if $k$ is even} c(g^{3/2}+1)(\Delta-1)^{\lfloor k/2\rfloor}
& \text{if $k$ is odd}, \{cases} where is an absolute constant. This
result represents a qualitative improvement over all previous results, even for
planar graphs of odd diameter . With respect to lower bounds, we construct
graphs of Euler genus , odd diameter , and order
for some absolute constant
. Our results answer in the negative a question of Miller and
\v{S}ir\'a\v{n} (2005).Comment: 13 pages, 3 figure
La forma lògica de les oracions d'acció i la tesi d'Anscombe
This paper has three sections. In the first one, I expose and discuss Davidson's semantic account of adverbial sentences: the basic idea is that these sentences involve quantification over events, and I defend that view from opposing perspectives like the theory of adverbs as predicate modifiers. In the second section I defend the claim that in english constructions following the scheme: "X did V by T-ings", we are referring to the same action of X; what is sometimes called "The Anscombe Thesis". Again I discuss competing theories only to conclude that the Anscombe Thesis is true. In the third section, however, it is shown that to assume as premisses these two theses -Davidson's account and the Anscombe Thesis- leads to a serious conflict. Alternative solutions are worked out and rejected. It is also argued that the only tenable solution depends on certain metaphysical assumptions. Finally, however, I will cast doubt on this solution
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