arXiv subjects

Rajat Adak

Publications and source records attributed to Rajat Adak.

14 recordsLinked to original sources

arXiv2026

Clique counts under degree and matching constraints

We study the maximum number of $t$-cliques in graphs with maximum degree at most $D$ and matching number at most $M$, with no restriction on the number of vertices. For $t\ge 3$, we determine this maximum whenever $D\ge 2M-1$ and characterize all extremal graphs when the maximum is positive. Up to isolated vertices, these are complete or complete split graphs for $D\ge 2M$, while $K_{2M}$ is uniquely extremal when $D=2M-1$. We also derive a general upper bound from the Chvátal-Hanson edge theorem and refine it for even $D$. For fixed even $D$ and $3\le t\le D+1$, every extremal graph consists, apart from isolated vertices, of copies of $K_{D+1}$ together with a remainder whose order is bounded independently of $M$. When $D/2$ divides $M$, this remainder is empty and the extremal graph is unique. This structure yields an eventual recurrence, reducing the determination of all extremal values and graphs to finitely many matching bounds.

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arXiv2026

Claw-Free and Line Graph Edge Deletion on Block Graphs

Edge deletion problems ask for the fewest edges whose deletion gives a graph with a specified property. We study two such problems on block graphs. First making the graph claw-free, and second making it a line graph. Since every line graph is claw-free, the second problem is at least as difficult as the first. We show that on block graphs the two optimum values are always equal, and both are attained by the line graph of a forest. The main step is a completion argument, in which starting from any claw-free solution, we put back edges to get some nice structure inside each block still maintaining the claw-free property. This reduces both edge-deletion problems to the same problem of selecting vertices within the blocks of the input graph. We solve it by dynamic programming in $O(n+m)$ time, which also produces an optimal deletion set and a forest whose line graph is what remains after the completion argument. We also extend the method to $K_{1,r+1}$-free edge deletion for every fixed $r$.

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arXiv2026

Linear Turán Numbers of Uniform Hypertrees

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 Gyárfás, Ruszinkó, and Sárközy from uniformity $3$ to general $r$, with emphasis on hypertrees with four hyperedges. For the linear path $P_4^r$, we prove $\operatorname{ex}^{\mathrm{lin}}_r(n,P_4^r)\le (r+1)n/r$, with equality precisely for disjoint unions of Steiner systems $S(2,r,r^2)$, whenever such systems exist. We also give counterexamples to a structural claim used in a previously proposed proof for $r=4$. The same bound and equality characterization hold for the broom $B_4^r$. For the crown $E_4^r$, writing $B=\{v:d(v)\le r\}$ and $C=\{v:r+1\le d(v)\le2r-1\}$, we prove $|E(H)|\le r|B|/(r-1)+(2r-1)|C|/r$, and hence $\operatorname{ex}^{\mathrm{lin}}_r(n,E_4^r)\le(2r-1)n/r$. We also give a lower-bound construction and formulate a sharper conjecture. Finally, for the linear star $S_k^r$, we prove $\operatorname{ex}^{\mathrm{lin}}_r(n,S_k^r)\le n(k-1)/r$, with equality precisely for $(k-1)$-regular linear $r$-uniform hypergraphs, whenever such hypergraphs exist. Under suitable divisibility and existence assumptions, we also construct $T_k^r$-free examples with $n(k-1)/r$ edges for every linear $r$-uniform hypertree $T_k^r$ with $k$ hyperedges.

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arXiv2026

On Eternal Connected Vertex Cover

For a connected graph $G$ with at least one edge, the \textit{eternal connected vertex cover} number $ecvc(G)$ is the minimum number of guards that can maintain a connected vertex cover after every response to an arbitrary sequence of edge attacks. Denote the minimum size of a connected vertex cover by $cvc(G)$. It is known that $cvc(G)\leq ecvc(G)\leq cvc(G)+1$. A necessary condition for $ecvc(G)=cvc(G)$ is that every vertex belongs to some minimum connected vertex cover. We show that this condition is not sufficient: there exists a $32$-vertex graph $G$ that has $cvc(G)=19$ and $ecvc(G)=20$, although every vertex of $G$ belongs to some minimum connected vertex cover. For connected graphs with minimum degree at least two, we establish the sharp bound $cvc(G)\geq 2|V(G)|-|E(G)|-1$ and $\mathcal F$ denotes the class attaining equality. We prove that $G\in\mathcal F$ if and only if the vertices of degree at least three induce a forest. Within $\mathcal F$, the conditions $ecvc(G)=cvc(G)$, membership of every vertex in some minimum connected vertex cover, and the presence of at least two degree-two vertices on every cycle are equivalent. As applications, we obtain $ecvc(G)=cvc(G)$ for full subdivisions of connected graphs of minimum degree at least two and for minimally $2$-connected graphs. In both the families, every minimum connected vertex cover is an eternally winning configuration. This is not true in general for graphs outside $\mathcal{F}$, we show one example of such a graph.

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arXiv2026

Generalized Zykov's Theorem

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 $K_{r+1}$-free graph and $t \geq 2$, \[ N(G,K_t) \leq \binom{r}{t}\left(\frac{n}{r}\right)^t. \] We generalize Zykov's bound within a vertex-based localization framework. For each vertex $v \in V(G)$, let $c(v)$ denote the order of the largest clique containing $v$. Then \[ N(G,K_t) \leq n^{t-1} \sum_{v \in V(G)} \frac{1}{c(v)^t}\binom{c(v)}{t}. \] Moreover, when $G$ contains a copy of $K_t$, equality holds if and only if $G$ is a regular complete multipartite graph. Note that if we impose the condition that $G$ is $K_{r+1}$-free, then $c(v) \leq r$ for all $v \in V(G)$, and the monotonicity of $s \mapsto \binom{s}{t}/s^t$ gives Zykov's bound.

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arXiv2026

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

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, one fixes a pair of linear equations $(\mathcal{E}_1, \mathcal{E}_2)$ and seeks the least integer $N$ such that every red--blue coloring of $\{1,2,\dots,N\}$ contains either a red solution to $\mathcal{E}_1$ or a blue solution to $\mathcal{E}_2$. This threshold integer is referred to as the off-diagonal Rado number of the system $(\mathcal{E}_1, \mathcal{E}_2)$. In this work, we study the discrete and continuous two-color off-diagonal Rado numbers for the nonhomogeneous linear equations $x+y+c=z$ and $x+y+k=z$, where $c\leq k$. In the discrete setting, $c$ and $k$ are nonnegative integers, whereas in the continuous setting, they are nonnegative real numbers. We determine the exact discrete and continuous two-color off-diagonal Rado numbers for this pair of shifted Schur equations.

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arXiv2026

Bounds on Linear Turán Number for Trees

A hypergraph $H$ is said to be \emph{linear} if every pair of vertices lies in at most one hyperedge. Given a family $\mathcal{F}$ of $r$-uniform hypergraphs, an $r$-uniform hypergraph $H$ is said to be \emph{$\mathcal{F}$-free} if it contains no member of $\mathcal{F}$ as a subhypergraph. The \emph{linear Turán number} $ex_r^{\mathrm{lin}}(n,\mathcal{F})$ denotes the maximum number of hyperedges in an $\mathcal{F}$-free linear $r$-uniform hypergraph on $n$ vertices. Gyárfás, Ruszinkó, and Sárközy~[\emph{Linear Turán numbers of acyclic triple systems}, European J.\ Combin.\ (2022)] initiated the study of bounds on the linear Turán number for acyclic $3$-uniform linear hypergraphs. In this paper, we extend the study of linear Turán numbers for acyclic systems to higher uniformity. We first give a construction for linear $r$-uniform trees with $k$ edges that yields the lower bound $ ex_r^{\mathrm{lin}}(n,T_k^r)\ge {n(k-1)}/{r}, $ under mild divisibility and existence assumptions. Next, we study hypertrees with four edges. We prove the exact bound $ ex_r^{\mathrm{lin}}(n,B_4^r)\le {(r+1)n}/{r} $ and characterize the extremal hypergraph class, where $B_4^r$ is formed from $S_3^r$ by appending a hyperedge incident to a degree-one vertex. We also prove the bound $ ex_r^{\mathrm{lin}}(n,E_4^r)\le {(2r-1)n}/{r} $ for the crown $E_4^r$. Finally, we give a construction showing $ ex_r^{\mathrm{lin}}(n,P_4^r)\ge {(r+1)n}/{r} $ under suitable assumptions and conclude with a conjecture on sharp upper bound for $P_4^r$.

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arXiv2026

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

A hypergraph $H$ is said to be \emph{linear} if every pair of vertices lies in at most one hyperedge. Given a family $\mathcal{F}$ of $r$-uniform hypergraphs (also called $r$-graphs), an $r$-graph $H$ is said to be \emph{$\mathcal{F}$-free} if it contains no member of $\mathcal{F}$ as a subhypergraph. The \emph{linear Turán number} $ex_r^{\mathrm{lin}}(n,\mathcal{F})$ denotes the maximum number of edges in an $\mathcal{F}$-free linear $r$-graph on $n$ vertices. The crown is a linear $3$-graph obtained from three pairwise disjoint edges by adding an edge that intersects each of them in a distinct vertex. Recently, Gyárfás, Ruszinkó, and Sárközy~[\emph{Linear Turán numbers of acyclic triple systems}, European J.\ Combin.\ (2022)] initiated the study of bounds on the linear Turán number for acyclic $3$-uniform linear hypergraphs, including that of the crown. We extend the notion of a crown by defining a $k$-crown, denoted by $C_{1,k}^r$, to be a linear $r$-graph consisting of one base edge together with $k$ pairwise disjoint edges, each intersecting the base in a distinct vertex. In this paper, we establish an upper bound on $ex_r^{\mathrm{lin}}(n,C_{1,k}^r)$, which in particular improves the recent bound of Zhang, Broersma, and Wang~[\emph{Generalized Crowns in Linear $r$-Graphs}, Electron.\ J.\ Combin.\ (2025)] for all $r \geq 4$, without forbidding any auxiliary configuration. We also note that the cases $k\in\{1,2\}$ correspond to the short linear paths $P_2^r$ and $P_3^r$, and can be treated separately.

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arXiv2026

Localization: A Framework to Generalize Extremal Graph Problems

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 framework that refines such problems by assigning extremal quantities locally (to vertices or edges) and then aggregating them. This perspective not only recovers classical results but also leads to sharper bounds. A classical result states that a connected planar graph with a finite girth $g$ satisfies \begin{equation*} m \leq \frac{g}{g-2}(n-2) \end{equation*} Wood~\cite{wood} derived upper bounds on the number of $K_t$-cliques in graphs of bounded maximum degree, expressed in terms of both the number of vertices and the number of edges: \begin{align*} ex(n,K_t,K_{1,d+1}) \leq \frac{n}{d+1}\binom{d+1}{t} \\ mex(m,K_t,K_{1,d+1}) \leq \frac{m}{\binom{d+1}{2}}\binom{d+1}{t} \end{align*} More recently, Chakraborty and Chen~\cite{CHAKRABORTI2024103955} established a similar upper bound for graphs with bounded path length: \begin{equation*} mex(m,K_t,P_{r+1}) \leq \frac{m}{\binom{r}{2}}\binom{r}{t} \end{equation*} In this paper, we employ the localization framework to improve these bounds and provide structural characterizations of the extremal graphs attaining them.

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arXiv2026

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

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 considers two different linear equations $(\mathcal{E}_1,\mathcal{E}_2)$ and determines the minimum integer $N$ for which any red-blue coloring of $\{1,2,...,N\}$ forces either a red solution of the equation $\mathcal{E}_1$ or a blue solution of the equation $\mathcal{E}_2$. In this work, we study off-diagonal Rado numbers for non-homogeneous linear equations of the forms $x+y+c=z$ and $x+qy=z$. We determine the exact two-color off-diagonal Rado number $R_2(c,q)$ associated with this system of equations.

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arXiv2025

Vertex-Based Localization of Generalized Turán Problems

Let $\mathcal{F}$ be a family of graphs. A graph is called $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$. Generalized Turán problems aim to maximize the number of copies of a graph $H$ in an $n$-vertex $\mathcal{F}$-free graph. This maximum is denoted by $ex(n, H, \mathcal{F})$. When $H \cong K_2$, it is simply denoted by $ex(n,F)$. Erdős and Gallai established the bounds $ex(n, P_{k+1}) \leq \frac{n(k-1)}{2}$ and $ex(n, C_{\geq k+1}) \leq \frac{k(n-1)}{2}$. This was later extended by Luo \cite{luo2018maximum}, who showed that $ex(n, K_s, P_{k+1}) \leq \frac{n}{k} \binom{k}{s}$ and $ex(n, K_s, C_{\geq k+1}) \leq \frac{n-1}{k-1} \binom{k}{s}$. Let $N(G,K_s)$ denote the number of copies of $K_s$ in $G$. In this paper, we use the vertex-based localization framework, introduced in \cite{adak2025vertex}, to generalize Luo's bounds. In a graph $G$, for each $v \in V(G)$, define $p(v)$ to be the length of the longest path that contains $v$. We show that \[N(G,K_s) \leq \sum_{v \in V(G)} \frac{1}{p(v)+1}{p(v)+1\choose s} = \frac{1}{s}\sum_{v \in V(G)}{p(v) \choose s-1}\] We strengthen the cycle bound from \cite{luo2018maximum} as follows: In graph $G$, for each $v \in V(G)$, let $c(v)$ be the length of the longest cycle that contains $v$, or $2$ if $v$ is not part of any cycle. We prove that \[N(G,K_s) \leq \left(\sum_{v\in V(G)}\frac{1}{c(v)-1}{c(v) \choose s}\right) - \frac{1}{c(u)-1}{c(u) \choose s}\] where $c(u)$ denotes the circumference of $G$. Furthermore, we characterize the class of extremal graphs that attain equality for these bounds. We provide full proofs for the cases $s = 1$ and $s \geq 3$, while the case $s = 2$ follows from the result in \cite{adak2025vertex}. We also conclude with a generalization of a result by Balister-Bollobás-Riordan-Schelp \cite{BALISTER2003366}.

math.CO↗

arXiv2025

Vertex-Based Localization of Turán's Theorem

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 graph on $n$ vertices with a maximum clique of order $r$. A limitation of this statement is that it does not give an expression in terms of $n$ and $r$. A widely used version of Turán's theorem states that for an $n$-vertex $K_{r+1}$-free graph, $m \leq \left\lfloor \frac{n^2(r-1)}{2r} \right\rfloor.$ Though this bound is often more convenient, it is not the same as the original statement. In particular, the class of extremal graphs for this bound, say $\mathcal{S}$, is a proper subset of the set of Turán graphs. In this paper, we generalize this result as follows: For each $v \in V(G)$, let $c(v)$ be the order of the largest clique that contains $v$. We show that \[ m \leq \left\lfloor\frac{n}{2}\sum_{v\in V(G)}\frac{c(v)-1}{c(v)}\right\rfloor\] Furthermore, we characterize the class of extremal graphs that attain equality in this bound. Interestingly, this class contains two extra non-Turán graphs other than the graphs in $\mathcal{S}$.

math.CO↗

arXiv2025

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

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, $m \leq \frac{n(k-1)}{2}$, where $P_k$ denotes a path with length $k$ (that is, with $k$ edges). In this paper, we generalize this result as follows: For each $v \in V(G)$, let $p(v)$ be the length of the longest path that contains $v$. We show that \[m \leq \sum_{v \in V(G)} \frac{p(v)}{2}\] The Erdős-Gallai theorem for cycles states that in a simple graph $G$ with circumference (that is, the length of the longest cycle) at most $k$, we have $m \leq \frac{k(n-1)}{2}$. We strengthen this result as follows: For each $v \in V(G)$, let $c(v)$ be the length of the longest cycle that contains $v$, or $2$ if $v$ is not part of any cycle. We prove that \[m \leq \left( \sum_{v \in V(G)} \frac{c(v)}{2} \right) - \frac{c(u)}{2}\] where $c(u)$ denotes the circumference of $G$. \newline Furthermore, we characterize the class of extremal graphs that attain equality in these bounds.

math.CO↗

arXiv2025

Revisiting Token Sliding on Chordal Graphs

In this article, we revisit the complexity of the reconfiguration of independent sets under the token sliding rule on chordal graphs. In the \textsc{Token Sliding-Connectivity} problem, the input is a graph $G$ and an integer $k$, and the objective is to determine whether the reconfiguration graph $TS_k(G)$ of $G$ is connected. The vertices of $TS_k(G)$ are $k$-independent sets of $G$, and two vertices are adjacent if and only if one can transform one of the two corresponding independent sets into the other by sliding a vertex (also called a \emph{token}) along an edge. Bonamy and Bousquet [WG'17] proved that the \textsc{Token Sliding-Connectivity} problem is polynomial-time solvable on interval graphs but \NP-hard on split graphs. In light of these two results, the authors asked: can we decide the connectivity of $TS_k(G)$ in polynomial time for chordal graphs with \emph{maximum clique-tree degree} $d$? We answer this question in the negative and prove that the problem is \para-\NP-hard when parameterized by $d$. More precisely, the problem is \NP-hard even when $d = 4$. We then study the parameterized complexity of the problem for a larger parameter called \emph{leafage} and prove that the problem is \co-\W[1]-hard. We prove similar results for a closely related problem called \textsc{Token Sliding-Reachability}. In this problem, the input is a graph $G$ with two of its $k$-independent sets $I$ and $J$, and the objective is to determine whether there is a sequence of valid token sliding moves that transform $I$ into $J$.

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