Varun Sivashankar

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20212026

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9 papers · 1 filter

math.CO2026

Nearly Spanning Regular Subgraphs

Varun Sivashankar

Alon and Mubayi asked whether, for every integer $k \geq 1$ and every $ε > 0$ , there exists $r_{0} = r_{0} (k, ε)$ such that every $r$ -regular graph on $n$ vertices with $r\…

math.CO2026

Zero-Sum Cycles in Regular Digraphs

Varun Sivashankar

Let $Γ$ be a finite group of order $k \geq 2$ , and label the edges of a simple loopless $d$ -regular digraph $D$ by elements of $Γ$ . A directed cycle is zero-sum if the ordered product…

math.CO2026

Graph Eigenvalues and Projection Constants

Varun Sivashankar, Quanyu Tang, Tanay Wakhare

For an integer $k \geq 2$ , let $λ_{k} (G)$ denote the $k$ th largest adjacency eigenvalue of a graph $G$ . For every graph $G$ on $n$ vertices and every $2 \leq k \leq n$ , we prove \[ λ_k(…

cs.GT2026

A Linear Bound on the Rainbow Cycle Number and Approximate EFX

Varun Sivashankar

It is open whether every fair-division instance with additive valuations admits a complete envy-free-up-to-any-good (EFX) allocation. A well-studied relaxation allows some goods to…

math.CO2026

An Improved Lower Bound for the Erdős-Lovász Cover Number Problem

Varun Sivashankar

Let $g (r)$ be the minimum number of edges in an $r$ -uniform intersecting hypergraph with cover number $r$ . Erdős and Lovász proved the lower bound $g (r) \geq 8 r /3 - 3$ . We first give a…

math.CO2026

Pseudoshattering Pairs

Noga Alon, Varun Sivashankar

For two vectors $x, y \in [b]^{k}$ , consider the bipartite graph with two copies of $[b]$ in which $i$ on the left is joined to $j$ on the right if $(x_{t}, y_{t}) = (i, j)$ for some coordinat…

math.CO2026

Tree-independence number and forbidden induced subgraphs: excluding a $6$ -vertex path and a $(2, t)$ -biclique

Maria Chudnovsky, Julien Codsi, J. Pascal Gollin +2

We show that for every positive integer $t \geq 2$ there exists an integer $s$ such that every graph that contains no induced subgraph isomorphic to either the $6$ -vertex path or…

math.CO2026

Upper bound on the $k$ -th eigenvalue of a graph

Varun Sivashankar

We prove a general upper bound on the $k$ -th adjacency eigenvalue of a graph. For $k \geq 2$ , we show that \[ λ_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1 \] for every graph $G$ …

math.CO2026

Extremal Graphs for the Lights Out Problem

Julien Codsi, Sergio Cristancho, Alexander Divoux +1

Lights Out is a game played on a graph $G$ where every vertex has a light bulb that is either on or off, and pressing a vertex $v$ toggles the state of every vertex in the closed n…

Nearly Spanning Regular Subgraphs

Zero-Sum Cycles in Regular Digraphs

Graph Eigenvalues and Projection Constants

A Linear Bound on the Rainbow Cycle Number and Approximate EFX

An Improved Lower Bound for the Erdős-Lovász Cover Number Problem

Pseudoshattering Pairs

Tree-independence number and forbidden induced subgraphs: excluding a 6-vertex path and a (2,t)-biclique

Upper bound on the k-th eigenvalue of a graph

Extremal Graphs for the Lights Out Problem

Tree-independence number and forbidden induced subgraphs: excluding a $6$ -vertex path and a $(2, t)$ -biclique

Upper bound on the $k$ -th eigenvalue of a graph