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S Selvaraja

Publications and source records attributed to S Selvaraja.

9 recordsLinked to original sources
arXiv2026

Regularity of Squarefree Powers of Edge Ideals of Whiskered Cycles

Let $G$ be a finite simple graph and let $I(G)$ denote its edge ideal. For $q \ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ is generated by squarefree monomials corresponding to matchings of size $q$ in $G$. We denote by $\operatorname{reg}(-)$ the Castelnuovo-Mumford regularity. Das, Roy, and Saha conjectured that if $G = W(C_n)$ is a whiskered cycle, then \[ \operatorname{reg}\big(I(G)^{[q]}\big) = 2q + \left\lfloor \frac{n - q - 1}{2} \right\rfloor ~ \text{for all } 1 \le q \le \nu(G), \] where $\nu(G)$ denotes the matching number of $G$. In this paper, we confirm this conjecture by determining the exact value of $\operatorname{reg}(I(G)^{[q]})$.

math.AC↗
arXiv2026

Cohen-Macaulayness of squarefree powers of edge ideals of whisker graphs

Let $G$ be a finite simple graph with edge ideal $I(G)$. For $q\ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ is generated by products of $q$ pairwise disjoint edges of $G$. It is the Stanley-Reisner ideal of a simplicial complex $\mathsf{MF}^q(G)$, called the $q$-matching-free complex, whose faces are those subsets $F\subseteq V(G)$ for which the induced subgraph $G[F]$ contains no matching of size $q$. We study $\mathsf{MF}^q(G)$ when $G=W(H)$ is a whisker graph. We first characterize purity. If $H$ is bipartite, then $\mathsf{MF}^q(G)$ is pure for all $q$. Otherwise, let $\ell$ denote the length of the smallest odd cycle of $H$ and set $n=|V(H)|$. Then $\mathsf{MF}^q(G)$ is pure if and only if $q<\lceil \ell/2\rceil$ or $q>n-\lfloor \ell/2\rfloor.$ We next determine the exact range of shellability. Let $m=\operatorname{girth}(H)$, with $m=\infty$ if $H$ is acyclic. Then $\mathsf{MF}^q(G)$ is shellable for \[ 1\le q\le \begin{cases} \lceil m/2\rceil, & \text{if } m<\infty,\\ \nu(G), & \text{if } m=\infty. \end{cases} \] Consequently, $I(G)^{[q]}$ is Cohen-Macaulay for $1\le q\le\lfloor m/2\rfloor$ when $m<\infty$, and for all $1\le q\le\nu(G)$ when $m=\infty$. If $m$ is odd, then $I(G)^{[q]}$ is sequentially Cohen-Macaulay for $q=\lceil m/2\rceil$. We further obtain extremal characterizations: $\mathsf{MF}^{2}(G)$ is Cohen-Macaulay if and only if $H$ has no induced $3$-cycle, and $\mathsf{MF}^{\,n-1}(G)$ is Cohen-Macaulay if and only if $H$ is acyclic. Finally, we compute the depth of $I(G)^{[q]}$ for whisker graphs and verify a conjecture on the depth of squarefree powers of whisker cycles in the relevant range.

math.AC↗
arXiv2026

Shellability in Clique-Free Complexes of Graphs

We study combinatorial and algebraic properties of $t$-clique-free complexes, a family of simplicial complexes associated with finite simple graphs that generalize the classical independence complex. For a graph $G$ and an integer $t \ge 2$, the $t$-clique-free complex $\mathsf{CF}_t(G)$ is the simplicial complex on the vertex set of $G$ whose faces are the subsets inducing no cliques of size $t$. Our main results provide sufficient conditions for shellability and related decomposability properties of $t$-clique-free complexes. In particular, we show that if $G$ is a $t$-diamond-free chordal graph (in particular, a block graph), then $\mathsf{CF}_t(G)$ is $(t-2)$-decomposable and hence shellable. We also investigate how graph modifications via clique attachments influence shellability. Generalizing earlier constructions involving whiskers and clique extensions, we introduce the following operation: given a graph $H$, a subset $S \subseteq V(H)$, and an integer $t \ge 2$, we form a graph $\operatorname{Cl}(H,S,t)$ by attaching to each vertex in $S$ a clique of size at least $t$. We prove that $\mathsf{CF}_t(H\setminus S)$ is shellable if and only if $\mathsf{CF}_t(\operatorname{Cl}(H,S,t))$ is shellable. This yields a flexible method for constructing shellable complexes, particularly when $S$ is a cycle cover. In addition, we extend the notion of clique whiskering and show that for any graph admitting a clique vertex-partition, the resulting $t$-clique whiskering produces a pure and shellable, and hence Cohen-Macaulay, $t$-clique-free complex. Finally, we establish a Fr\"oberg-type result linking chordality and linear resolutions. We show that for any chordal graph $G$, the edge ideal of the complement $t$-clique clutter $\overline{\mathcal{CH}_t(G)}$ admits a $t$-linear resolution over any field.

math.CO↗
arXiv2025

Linear resolution of connected graph ideals and their powers

For a finite simple graph $G$ and an integer $r \ge 1$, the $r$-connected ideal $I_r(G)$ is the squarefree monomial ideal generated by the vertex sets of connected induced subgraphs of size $r+1$, extending the classical edge ideal. We investigate the linearity of the minimal free resolutions of $I_r(G)$ via structural features of the associated clutter $\mathcal{C}_r(G)$. We introduce the class of co-chordal-cactus graphs and prove that $I_r(G)$ has a linear resolution for all $r \ge 2$ whenever $G$ lies in this family. The result further extends to $(2K_2, C_4)$-free graphs and co-grid graphs. For $r=1$, we show that the edge ideal $I_1(G)$ has Castelnuovo-Mumford regularity at most $3$ for all co-chordal-cactus and co-grid graphs. We also examine powers of connected ideals and establish that $I_r(G)^q$ has a linear resolution for every $q \ge 1$ in several natural graph families, including complements of trees with bounded degree, complete multipartite graphs, complements of cycles, graphs obtained by gluing complete graphs along cliques, and certain subclasses of split graphs.

math.AC↗
arXiv2025

Shellability of Higher Independence Complexes of Graphs

This paper investigates the shellability of $r$-independence complexes $\mathcal{I}_r(G)$, a generalization of classical independence complexes introduced by Paolini and Salvetti. For a graph $G$, a subset $A \subseteq V(G)$ is $r$-independent if every connected component of the induced subgraph $G[A]$ has at most $r$ vertices. The associated simplicial complex $\mathcal{I}_r(G)$ has been the subject of significant interest due to its connections to combinatorial topology and commutative algebra. We address the classification problem for shellable $r$-independence complexes, focusing on block graphs, trees, and related families. Our main results establish sufficient conditions for shellability based on structural graph parameters such as diameter and forbidden subgraphs. Furthermore, we develop constructive techniques for generating shellable complexes through graph operations, including star-clique attachments, clique whiskering, and clique cycle constructions. These results extend and refine earlier work on classical independence complexes and provide a framework for understanding the topological and algebraic properties of higher independence complexes in structured graph families.

math.CO↗
arXiv2021

Upper bounds for the regularity of symbolic powers of certain classes of edge ideals

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of symbolic powers of certain classes of edge ideals. We also prove that for several classes of graphs, the regularity of symbolic powers of their edge ideals coincides with that of their ordinary powers.

math.AC↗
arXiv2019

Symbolic powers of vertex cover ideals

Let $G$ be a finite simple graph and $J(G)$ denote its cover ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we show that all symbolic powers of cover ideals of certain vertex decomposable graphs have linear quotients. Using these results, we give various conditions on a subset $S$ of the vertices of $G$ so that all symbolic powers of vertex cover ideals of $G \cup W(S)$, obtained from $G$ by adding a whisker to each vertex in $S$, have linear quotients. For instance, if $S$ is a vertex cover of $G$, then all symbolic powers of $J(G \cup W(S))$ have linear quotients. Moreover, we compute the Castelnuovo-Mumford regularity of symbolic powers of certain cover ideals.

math.AC↗
arXiv2016

Regularity of Powers of Bipartite Graphs

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. For all $s \geq 1$, we obtain upper bounds for reg$(I(G)^s)$ for bipartite graphs. We then compare the properties of $G$ and $G'$, where $G'$ is the graph associated with the polarization of the ideal $(I(G)^{s+1} : e_1\cdots e_s)$, where $e_1,\ldots e_s$ are edges of $G$. Using these results, we explicitly compute reg$(I(G)^s)$ for several subclasses of bipartite graphs.

math.AC↗