30,544 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
A necessary and sufficient condition for lower bounds on crossing numbers of generalized periodic graphs in an arbitrary surface
Let , and be a graph, a tree and a cycle of order ,
respectively. Let be the complete join of and an empty graph on
vertices. Then the Cartesian product of and can be
obtained by applying zip product on and the graph produced by zip
product repeatedly. Let denote the crossing number of
in an arbitrary surface . If satisfies certain connectivity
condition, then is not less than the sum of the
crossing numbers of its ``subgraphs". In this paper, we introduced a new
concept of generalized periodic graphs, which contains . For a
generalized periodic graph and a function , where is the number
of subgraphs in a decomposition of , we gave a necessary and sufficient
condition for . As an application, we
confirmed a conjecture of Lin et al. on the crossing number of the generalized
Petersen graph in the plane. Based on the condition, algorithms
are constructed to compute lower bounds on the crossing number of generalized
periodic graphs in . In special cases, it is possible to determine
lower bounds on an infinite family of generalized periodic graphs, by
determining a lower bound on the crossing number of a finite generalized
periodic graph.Comment: 26 pages, 20 figure
Bar 1-Visibility Drawings of 1-Planar Graphs
A bar 1-visibility drawing of a graph is a drawing of where each
vertex is drawn as a horizontal line segment called a bar, each edge is drawn
as a vertical line segment where the vertical line segment representing an edge
must connect the horizontal line segments representing the end vertices and a
vertical line segment corresponding to an edge intersects at most one bar which
is not an end point of the edge. A graph is bar 1-visible if has a bar
1-visibility drawing. A graph is 1-planar if has a drawing in a
2-dimensional plane such that an edge crosses at most one other edge. In this
paper we give linear-time algorithms to find bar 1-visibility drawings of
diagonal grid graphs and maximal outer 1-planar graphs. We also show that
recursive quadrangle 1-planar graphs and pseudo double wheel 1-planar graphs
are bar 1-visible graphs.Comment: 15 pages, 9 figure
Tropical Hurwitz Numbers
Hurwitz numbers count genus g, degree d covers of the projective line with
fixed branch locus. This equals the degree of a natural branch map defined on
the Hurwitz space. In tropical geometry, algebraic curves are replaced by
certain piece-wise linear objects called tropical curves. This paper develops a
tropical counterpart of the branch map and shows that its degree recovers
classical Hurwitz numbers.Comment: Published in Journal of Algebraic Combinatorics, Volume 32, Number 2
/ September, 2010. Added section on genus zero piecewise polynomiality.
Removed paragraph on psi classe
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