617 research outputs found
The crossing numbers of join of the special graph on six vertices with path and cycle
AbstractThere are only few results concerning crossing numbers of join of some graphs. In the paper, for the special graph H on six vertices we give the crossing numbers of its join with n isolated vertices as well as with the path Pn on n vertices and with the cycle Cn
The Crossing Numbers of Cartesian Products of Stars with 5-Vertex Graphs
In this paper, the crossing number of Cartesian products of a speci c 5-vertex graph with a star are given, and this lls up the crossing number list of Cartesian products of all 5-vertex graphs with stars (presented by Marian Klesc)
The Crossing Number of Two Cartesian Products
There are several known exact results on the crossing number of Cartesian
products of paths, cycles, and complete graphs. In this paper, we find the crossing numbers of Cartesian products of Pn with two special 6-vertex graphs
Hypercellular graphs: partial cubes without as partial cube minor
We investigate the structure of isometric subgraphs of hypercubes (i.e.,
partial cubes) which do not contain finite convex subgraphs contractible to the
3-cube minus one vertex (here contraction means contracting the edges
corresponding to the same coordinate of the hypercube). Extending similar
results for median and cellular graphs, we show that the convex hull of an
isometric cycle of such a graph is gated and isomorphic to the Cartesian
product of edges and even cycles. Furthermore, we show that our graphs are
exactly the class of partial cubes in which any finite convex subgraph can be
obtained from the Cartesian products of edges and even cycles via successive
gated amalgams. This decomposition result enables us to establish a variety of
results. In particular, it yields that our class of graphs generalizes median
and cellular graphs, which motivates naming our graphs hypercellular.
Furthermore, we show that hypercellular graphs are tope graphs of zonotopal
complexes of oriented matroids. Finally, we characterize hypercellular graphs
as being median-cell -- a property naturally generalizing the notion of median
graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier
draft (Figure 6.
Coherence for indexed symmetric monoidal categories
Indexed symmetric monoidal categories are an important refinement of
bicategories -- this structure underlies several familiar bicategories,
including the homotopy bicategory of parametrized spectra, and its equivariant
and fiberwise generalizations.
In this paper, we extend existing coherence theorems to the setting of
indexed symmetric monoidal categories. The most central theorem states that a
large family of operations on a bicategory defined from an indexed symmetric
monoidal category are all canonically isomorphic. As a part of this theorem, we
introduce a rigorous graphical calculus that specifies when two such operations
admit a canonical isomorphism.Comment: 100 pages, 64 figures, 13 table
International Journal of Mathematical Combinatorics, Vol.1
The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences
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