arXiv subjects

Pratiksha Chauhan

Publications and source records attributed to Pratiksha Chauhan.

4 recordsLinked to original sources

arXiv2026

Homology of matching complexes of $3\times n$ grid graphs

For a finite simple graph $G$, the matching complex $M(G)$ is the simplicial complex whose vertex set is the edge set of $G$ and whose simplices are all the matchings in $G$. The topology of the matching complex of the $m\times n$ grid graph $G_{m\times n}$ is known only for $m = 1,2$, in which cases it is homotopy equivalent to a wedge of spheres. In this article, we study the matching complex $M(G_{3 \times n})$. We prove that for $n\ge2$, its reduced homology vanishes in dimensions $i \leq n-2$ and in top dimension, while $\tilde{H}_{n-1}(M(G_{3\times n}))\neq 0$. We also show that $M(G_{3 \times n})$ is simply connected for $n \geq 3$. Consequently, the topological connectivity of $M(G_{3\times n})$ is $n-2$.

math.CO↗

arXiv2026

Shellability of 3-cut complexes of powers of cycle graphs

In connection with commutative algebra, Bayer et al. introduced cut complexes in [Topology of cut complexes of graphs, SIAM J.\ Discrete Math., 38(2):1630-1675, 2024]. For a positive integer $k$, the $k$-cut complex of a graph $G$, denoted as $Δ_k(G)$, is the simplicial complex whose facets are the $(|V(G)|-k)$-subsets $σ$ of the vertex set $V(G)$ of $G$ such that the induced subgraph $G[V(G) \setminus σ]$ is disconnected. Let $C_n^p$ denote the $p$-th power graph of the cycle graph $C_n$ on $n$ vertices. In this article, we show that $Δ_3(C_n^p)$ is shellable for $n \geq 6p-3$, and therefore these complexes are homotopy equivalent to a wedge of spheres of dimension $n-4$. We provide an explicit shelling order on the facets of $Δ_3(C_n^p)$. We also characterize and count the number of spanning facets in this shelling order, and determine the number of spheres appearing in the wedge in the homotopy type of $Δ_3(C_n^p)$.

math.CO↗

arXiv2025

Total $2$-cut complexes of powers of cycle graphs and Cartesian products of certain graphs

For a positive integer $k$, the \emph{ total $k$-cut complex} of a graph $G$, denoted as $Δ_k^t(G)$, is the simplicial complex whose facets are $σ\subseteq V(G)$ such that $|σ| = |V(G)|-k$ and the induced subgraph $G[V(G) \setminus σ]$ does not contain any edge. These complexes were introduced by Bayer et al.\ in \cite{Bayer2024TotalCutcomplex} in connection with commutative algebra. In the same paper, they studied the homotopy types of these complexes for various families of graphs, including cycle graphs $C_n$, squared cycle graphs $C_n^2$, and Cartesian products of complete graphs and path graphs $K_m \square P_2$ and $K_2 \square P_n$. In this article, we extend the work of Bayer et al.\ for these families of graphs. We focus on the complexes $Δ_2^t(G)$ and determine the homotopy types of these complexes for three classes of graphs: (i) $p$-th powers of cycle graphs $C_n^p$ (ii) $K_m \square P_n$ and (iii) $K_m \square C_n$. Using discrete Morse theory, we show that these complexes are homotopy equivalent to wedges of spheres. We also give the number and dimension of spheres appearing in the homotopy type. Our result on powers of cycle graphs $C_n^p$ proves a conjecture of Shen et al.\ about the homotopy type of the complexes $Δ_2^t(C_n^p)$.

math.CO↗

arXiv2025

Shellability of $3$-Cut Complexes of Squared Cycle Graphs

For a positive integer $k$, the $k$-cut complex of a graph $G$ is the simplicial complex whose facets are the $(|V(G)|-k)$-subsets $σ$ of the vertex set $V(G)$ of $G$ such that the induced subgraph of $G$ on $V(G) \setminus σ$ is disconnected. These complexes first appeared in the master thesis of Denker and were further studied by Bayer et al.\ in [Topology of cut complexes of graphs, SIAM Journal on Discrete Mathematics, 2024]. In the same article, Bayer et al.\ conjectured that for $k \geq 3$, the $k$-cut complexes of squared cycle graphs are shellable. Moreover, they also conjectured about the Betti numbers of these complexes when $k=3$. In this article, we prove these conjectures for $k=3$.

math.CO↗