5,534 research outputs found
A Note on the Maximum Genus of Graphs with Diameter 4
Let G be a simple graph with diameter four, if G does not contain complete
subgraph K3 of order three
LR characterization of chirotopes of finite planar families of pairwise disjoint convex bodies
We extend the classical LR characterization of chirotopes of finite planar
families of points to chirotopes of finite planar families of pairwise disjoint
convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a
chirotope of finite planar families of pairwise disjoint convex bodies if and
only if for every 3-, 4-, and 5-subset J of I the restriction of \c{hi} to the
set of 3-subsets of J is a chirotope of finite planar families of pairwise
disjoint convex bodies. Our main tool is the polarity map, i.e., the map that
assigns to a convex body the set of lines missing its interior, from which we
derive the key notion of arrangements of double pseudolines, introduced for the
first time in this paper.Comment: 100 pages, 73 figures; accepted manuscript versio
Genus Ranges of 4-Regular Rigid Vertex Graphs
We introduce a notion of genus range as a set of values of genera over all
surfaces into which a graph is embedded cellularly, and we study the genus
ranges of a special family of four-regular graphs with rigid vertices that has
been used in modeling homologous DNA recombination. We show that the genus
ranges are sets of consecutive integers. For any positive integer , there
are graphs with vertices that have genus range for all
, and there are graphs with vertices with genus range
for all or . Further, we show that
for every there is such that is a genus range for graphs with
and vertices for all . It is also shown that for every ,
there is a graph with vertices with genus range , but there
is no such a graph with vertices
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