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 $k$-cut complex of a graph $G$ with vertex set $V(G)$ to be the simplicial complex whose facets are the complements of sets of size $k$ in $V(G)$ inducing disconnected subgraphs of $G$. 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 $K_n \times K_2$, 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 $K_{a,b,n}$, where $a,b\leq 2$.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 $K_{m,n,p}$ and the exact value for $K_{2,2,n}$. 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