Rajat Adak

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11 papers · 1 filter

math.CO2026

On Eternal Connected Vertex Cover

Rajat Adak, Saraswati Girish Nanoti

For a connected graph $G$ with at least one edge, the \textit{eternal connected vertex cover} number $ec v c (G)$ is the minimum number of guards that can maintain a connected vertex…

math.CO2026

Linear Turán Numbers of Uniform Hypertrees

Rajat Adak, Pragya Verma

A hypergraph is linear if every pair of vertices is contained in at most one hyperedge. We study linear Turán numbers of acyclic $r$ -uniform hypergraphs, extending earlier work of…

math.CO2026

An Upper Bound on the Linear Turán Number of $k$ -Crowns

Rajat Adak

A hypergraph $H$ is said to be \emph{linear} if every pair of vertices lies in at most one hyperedge. Given a family $F$ of $r$ -uniform hypergraphs (also called $r$ -graph…

math.CO2026

Off-diagonal Rado numbers for $x + y + c = z$ and $x + y + k = z$

Rajat Adak, Yash Bakshi, L. Sunil Chandran +1

The study of Ramsey-type problems for linear equations originated with Schur's theorem and was later placed in a systematic framework by Richard Rado. In the off-diagonal setting,…

math.CO2026

Off-diagonal Rado number for $x + y + c = z$ and $x + q y = z$

Rajat Adak, Yash Bakshi, L. Sunil Chandran +1

Ramsey-type problems for linear equations began with Schur's theorem and were systematically generalized by Richard Rado. In the off-diagonal framework for two colors, one consider…

math.CO2026

Bounds on Linear Turán Number for Trees

Rajat Adak, Pragya Verma

A hypergraph $H$ is said to be \emph{linear} if every pair of vertices lies in at most one hyperedge. Given a family $F$ of $r$ -uniform hypergraphs, an $r$ -uniform hyperg…

math.CO2025

Generalized Zykov's Theorem

Rajat Adak, L. Sunil Chandran

For a simple graph $G$ , let $n$ denote its number of vertices, and let $N (G, K_{t})$ denote the number of copies of $K_{t}$ in $G$ . Zykov's theorem (1949) asserts that for any $K_{r + 1}$ …

math.CO2025

Vertex-Based Localization of Generalized Turán Problems

Rajat Adak, L. Sunil Chandran

Let $F$ be a family of graphs. A graph is called $F$ -free if it does not contain any member of $F$ . Generalized Turán problems aim to maximize the num…

math.CO2025

Localization: A Framework to Generalize Extremal Graph Problems

Rajat Adak, L. Sunil Chandran

Extremal graph theory studies the maximum or minimum number of subgraphs isomorphic to a prescribed graph under given constraints. \textit{Localization} has recently emerged as a f…

math.CO2025

Vertex-Based Localization of Turán's Theorem

Rajat Adak, L. Sunil Chandran

Let $G$ be a simple graph with $n$ vertices and $m$ edges. According to Turán's theorem, if $G$ is $K_{r + 1}$ -free, then $m \leq ∣ E (T (n, r)) ∣,$ where $T (n, r)$ denotes the Turán gra…

math.CO2025

Vertex-Based Localization of Erdős-Gallai Theorems for Paths and Cycles

Rajat Adak, L. Sunil Chandran

For a simple graph $G$ , let $n$ and $m$ denote the number of vertices and edges in $G$ , respectively. The Erdős-Gallai theorem for paths states that in a simple $P_{k}$ -free graph, $…

On Eternal Connected Vertex Cover

Linear Turán Numbers of Uniform Hypertrees

An Upper Bound on the Linear Turán Number of k-Crowns

Off-diagonal Rado numbers for x+y+c=z and x+y+k=z

Off-diagonal Rado number for x+y+c=z and x+qy=z

Bounds on Linear Turán Number for Trees

Generalized Zykov's Theorem

Vertex-Based Localization of Generalized Turán Problems

Localization: A Framework to Generalize Extremal Graph Problems

Vertex-Based Localization of Turán's Theorem

Vertex-Based Localization of Erdős-Gallai Theorems for Paths and Cycles

An Upper Bound on the Linear Turán Number of $k$ -Crowns

Off-diagonal Rado numbers for $x + y + c = z$ and $x + y + k = z$

Off-diagonal Rado number for $x + y + c = z$ and $x + q y = z$