6 research outputs found
General polygonal line tilings and their matching complexes
A (general) polygonal line tiling is a graph formed by a string of cycles,
each intersecting the previous at an edge, no three intersecting. In 2022,
Matsushita proved the matching complex of a certain type of polygonal line
tiling with even cycles is homotopy equivalent to a wedge of spheres. In this
paper, we extend Matsushita's work to include a larger family of graphs and
carry out a closer analysis of lines of triangle and pentagons, where the
Fibonacci numbers arise.Comment: 22 page
Topology of Cut Complexes of Graphs
We define the -cut complex of a graph with vertex set to be the
simplicial complex whose facets are the complements of sets of size in
inducing disconnected subgraphs of . This generalizes the Alexander
dual of a graph complex studied by Fr\"oberg (1990), and Eagon and Reiner
(1998). We describe the effect of various graph operations on the cut complex,
and study its shellability, homotopy type and homology for various families of
graphs, including trees, cycles, complete multipartite graphs, and the prism
, using techniques from algebraic topology, discrete Morse
theory and equivariant poset topology.Comment: 36 pages, 10 figures, 1 table, Extended Abstract accepted for
FPSAC2023 (Davis
The tripartite-circle crossing number of graphs with two small partition classes
A tripartite-circle drawing of a tripartite graph is a drawing in the plane,
where each part of a vertex partition is placed on one of three disjoint
circles, and the edges do not cross the circles. The tripartite-circle crossing
number of a tripartite graph is the minimum number of edge crossings among all
its tripartite-circle drawings. We determine the exact value of the
tripartite-circle crossing number of , where .Comment: 22 pages, 11 figures. Added new results and revised throughout.
Originally appeared in arXiv:1910.06963v1, now removed from
arXiv:1910.06963v
Bounding the tripartite-circle crossing number of complete tripartite graphs
A tripartite-circle drawing of a tripartite graph is a drawing in the plane,
where each part of a vertex partition is placed on one of three disjoint
circles, and the edges do not cross the circles. We present upper and lower
bounds on the minimum number of crossings in tripartite-circle drawings of
and the exact value for . In contrast to 1- and 2-circle
drawings, which may attain the Harary-Hill bound, our results imply that
balanced restricted 3-circle drawings of the complete graph are not optimal