arXiv subjects

Debsoumya Chakraborti

Publications and source records attributed to Debsoumya Chakraborti.

At least 19 recordsLinked to original sources

arXiv2026

Sabotaging Mantel's Theorem

One of the earliest results in extremal graph theory, Mantel's theorem, states that the maximum number of edges in a triangle-free graph $G$ on $n$ vertices is $\lfloor n^2/4 \rfloor$. We investigate how this extremal bound is affected when $G$ is additionally required to contain a prescribed graph $H$ as a subgraph. We establish general upper and lower bounds for this problem, which are tight in the exponent for random triangle-free graphs and graphs generated by the triangle-free process, when the size of $H$ lies within certain ranges.

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arXiv2026

Large rainbow matchings in edge-colored graphs

A subgraph of an edge-colored graph is called \emph{rainbow} if all of its edges have distinct colors. There has been much research on the topic of finding a large rainbow matching in a properly edge-colored graph, where a proper edge-coloring is a coloring of the edge set such that no same-colored edges are incident. Gao, Ramadurai, Wanless, and Wormald proved that in every proper edge-coloring of a graph with $n$ colors where each color appears at least $n+o(n)$ times, there is always a rainbow matching using every color. We strengthen this result by simultaneously relaxing three conditions: (i) we lift the condition on the number of colors and allow any finite number of colors and instead, put a weaker condition requiring the maximum degree of the graph to be at most $n$, (ii) we relax the proper coloring condition and require that the graph induced by each of the colors have maximum degree $o(n)$, and (iii) we work in a more general setting of multigraphs allowing edge multiplicities to be $o(n)$. As an application of this result, we show that for every proper edge-coloring of a graph with $2n+o(n)$ colors where each color appears at least $n$ times, there is always a rainbow matching of size $n$. Aharoni and Berger conjectured that $2n+o(n)$ can be replaced by $n+1$ in this statement. We dispute this conjecture with an explicit construction.

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arXiv2026

Robust Hamiltonicity in families of Dirac graphs

A graph is called Dirac if its minimum degree is at least half of the number of vertices in it. Joos and Kim showed that every collection $\mathbb{G}=\{G_1,\ldots,G_n\}$ of Dirac graphs on the same vertex set $V$ of size $n$ contains a Hamilton cycle transversal, i.e., a Hamilton cycle $H$ on $V$ with a bijection $ϕ:E(H)\rightarrow [n]$ such that $e\in G_{ϕ(e)}$ for every $e\in E(H)$. In this paper, we determine up to a multiplicative constant, the threshold for the existence of a Hamilton cycle transversal in a collection of random subgraphs of Dirac graphs in various settings. Our proofs rely on constructing a spread measure on the set of Hamilton cycle transversals of a family of Dirac graphs. As a corollary, we obtain that every collection of $n$ Dirac graphs on $n$ vertices contains at least $(cn)^{2n}$ different Hamilton cycle transversals $(H,ϕ)$ for some absolute constant $c>0$. This is optimal up to the constant $c$. Finally, we show that if $n$ is sufficiently large, then every such collection spans $n/2$ pairwise edge-disjoint Hamilton cycle transversals, and this is best possible. These statements generalize classical counting results of Hamilton cycles in a single Dirac graph.

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arXiv2025

Regular subgraphs at every density

In 1975, Erdős and Sauer asked to estimate, for any constant $r$, the maximum number of edges an $n$-vertex graph can have without containing an $r$-regular subgraph. In a recent breakthrough, Janzer and Sudakov proved that any $n$-vertex graph with no $r$-regular subgraph has at most $C_r n \log \log n$ edges, matching an earlier lower bound by Pyber, Rödl and Szemerédi and thereby resolving the Erdős-Sauer problem up to a constant depending on $r$. We prove that every $n$-vertex graph without an $r$-regular subgraph has at most $Cr^2 n \log \log n$ edges. This bound is tight up to the value of $C$ for $n\geq n_0(r)$ and hence resolves the Erdős-Sauer problem up to an absolute constant. Moreover, we obtain similarly tight results for the whole range of possible values of $r$ (i.e., not just when $r$ is a constant), apart from a small error term at a transition point near $r\approx \log n$, where, perhaps surprisingly, the answer changes. More specifically, we show that every $n$-vertex graph with average degree at least $\min(Cr\log(n/r),Cr^2 \log\log n)$ contains an $r$-regular subgraph. The bound $Cr\log(n/r)$ is tight for $r\geq \log n$, while the bound $Cr^2 \log \log n$ is tight for $r<(\log n)^{1-Ω(1)}$. These results resolve a problem of Rödl and Wysocka from 1997 for almost all values of $r$. Among other tools, we develop a novel random process that efficiently finds a very nearly regular subgraph in any almost-regular graph. A key step in our proof uses this novel random process to show that every $K$-almost-regular graph with average degree $d$ contains an $r$-regular subgraph for some $r=Ω_K(d)$, which is of independent interest.

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arXiv2025

Colour-biased Hamilton cycles in dense graphs and random graphs

A classical result of Dirac says that every $n$-vertex graph with minimum degree at least $\frac{n}{2}$ contains a Hamilton cycle. A `discrepancy' version of Dirac's theorem was shown by Balogh--Csaba--Jing--Pluhár, Freschi--Hyde--Lada--Treglown, and Gishboliner--Krivelevich--Michaeli as follows. Every $r$-colouring of the edge set of every $n$-vertex graph with minimum degree at least $(\frac{1}{2} + \frac{1}{2r} + o(1))n$ contains a Hamilton cycle where one of the colours appears at least $(1+o(1))\frac{n}{r}$ times. In this paper, we generalize this result by asymptotically determining the maximum possible value $f_{r,α}(n)$ for every $α\in [\frac{1}{2}, 1]$ such that every $r$-colouring of the edge set of every $n$-vertex graph with minimum degree at least $αn$ contains a Hamilton cycle where one of the colours appears at least $f_{r,α}(n)$ times. In particular, we show that $f_{r,α}(n) = (1-o(1)) \min\{(2α- 1)n, \frac{2αn}{r}, \frac{2n}{r+1}\}$ for every $α\in [\frac{1}{2} + \frac{1}{2r}, 1]$. A graph $H$ is called an $α$-residual subgraph of a graph $G$ if $d_H(v)\ge αd_{G[V(H)]}(v)$ for every $v\in V(H)$. Extending Dirac's theorem in the setting of random graphs, Lee and Sudakov showed the following. The Erdős--Rényi random graph $G(n,p)$, with $p$ above the Hamiltonicity threshold, typically has the property that every $(\frac{1}{2} +o(1))$-residual spanning subgraph contains a Hamilton cycle. Motivated by this, we prove the following random version of our `discrepancy' result. The random graph $G \sim G(n,p)$, with $p$ above the Hamiltonicity threshold, typically satisfies that every $r$-colouring of the edge set of every $α$-residual spanning subgraph of $G$ contains a Hamilton cycle where one of the colours appears at least $f_{r,α}(n)$ times.

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arXiv2025

Approximate packing of independent transversals in locally sparse graphs

Fix $\varepsilon >0$ and consider a multipartite graph $G$ with maximum degree at most $(1-\varepsilon)n$, parts $V_1,\ldots,V_k$ of the same size $n$, and where every vertex has at most $o(n)$ neighbors in any part $V_i$. Loh and Sudakov proved that any such $G$ has an independent transversal. They further conjectured that the vertex set of $G$ can be decomposed into pairwise disjoint independent transversals. In the present paper, we resolve this conjecture approximately by showing that $G$ contains $(1-\varepsilon)n$ pairwise disjoint independent transversals. As applications, we give approximate answers to questions of Yuster, and of Fischer, Kühn, and Osthus.

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arXiv2024

Almost-full transversals in equi-$n$-squares

In 1975, Stein made a wide generalisation of the Ryser-Brualdi-Stein conjecture on transversals in Latin squares, conjecturing that every equi-$n$-square (an $n\times n$ array filled with $n$ symbols where each symbol appears exactly $n$ times) has a transversal of size $n-1$. That is, it should have a collection of $n-1$ entries that share no row, column, or symbol. In 2017, Aharoni, Berger, Kotlar, and Ziv showed that equi-$n$-squares always have a transversal with size at least $2n/3$. In 2019, Pokrovskiy and Sudakov disproved Stein's conjecture by constructing equi-$n$-squares without a transversal of size $n-\frac{\log n}{42}$, but asked whether Stein's conjecture is approximately true. I.e., does an equi-$n$-square always have a transversal with size $(1-o(1))n$? We answer this question in the positive. More specifically, we improve both known bounds, showing that there exist equi-$n$-squares with no transversal of size $n-Ω(\sqrt{n})$ and that every equi-$n$-square contains $n-n^{1-Ω(1)}$ disjoint transversals of size $n-n^{1-Ω(1)}$.

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arXiv2024

Sparse graphs with local covering conditions on edges

In 1988, Erdős suggested the question of minimizing the number of edges in a connected $n$-vertex graph where every edge is contained in a triangle. Shortly after, Catlin, Grossman, Hobbs, and Lai resolved this in a stronger form. In this paper, we study a natural generalization of the question of Erdős in which we replace `triangle' with `clique of order $k$' for ${k\ge 3}$. We completely resolve this generalized question with the characterization of all extremal graphs. Motivated by applications in data science, we also study another generalization of the question of Erdős where every edge is required to be in at least $\ell$ triangles for $\ell\ge 2$ instead of only one triangle. We completely resolve this problem for $\ell = 2$.

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arXiv2024

Colorful fractional Helly theorem via weak saturation

Two celebrated extensions of the classical Helly's theorem are the fractional Helly theorem and the colorful Helly theorem. Bulavka, Goodarzi, and Tancer recently established the optimal bound for the unified generalization of the fractional and the colorful Helly theorems using a colored extension of the exterior algebra. In this paper, we combinatorially reduce both the fractional Helly theorem and its colorful version to a classical problem in extremal combinatorics known as {weak saturation}. No such results connecting the fractional Helly theorem and weak saturation are known in the long history of literature. These reductions, along with basic linear algebraic arguments for the reduced weak saturation problems, let us give new short proofs of the optimal bounds for both the fractional Helly theorem and its colorful version without using exterior algebra.

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arXiv2024

Transversal cycles and paths in tournaments

Thomason [$\textit{Trans. Amer. Math. Soc.}$ 296.1 (1986)] proved that every sufficiently large tournament contains Hamilton paths and cycles with all possible orientations, except possibly the consistently oriented Hamilton cycle. This paper establishes $\textit{transversal}$ generalizations of these classical results. For a collection $\mathbf{T}=\{T_1,\dots,T_m\}$ of not-necessarily distinct tournaments on the common vertex set $V$, an $m$-edge directed subgraph $\mathcal{D}$ with the vertices in $V$ is called a transversal if there exists an bijection $φ\colon E(\mathcal{D})\to [m]$ such that $e\in E(T_{φ(e)})$ for all $e\in E(\mathcal{D})$. We prove that for sufficiently large $n$, there exist transversal Hamilton cycles of all possible orientations possibly except the consistently oriented one. We also obtain a similar result for the transversal Hamilton paths of all possible orientations. These results generalize the classical theorem of Thomason, and our approach provides another proof of this theorem.

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arXiv2024

Hamilton transversals in tournaments

It is well-known that every tournament contains a Hamilton path, and every strongly connected tournament contains a Hamilton cycle. This paper establishes transversal generalizations of these classical results. For a collection $\mathbf{T}=\{T_1,\dots,T_m\}$ of not-necessarily distinct tournaments on a common vertex set $V$, an $m$-edge directed graph $\mathcal{D}$ with vertices in $V$ is called a $\mathbf{T}$-transversal if there exists a bijection $ϕ\colon E(\mathcal{D})\to [m]$ such that $e\in E(T_{ϕ(e)})$ for all $e\in E(\mathcal{D})$. We prove that for sufficiently large $m$ with $m=|V|-1$, there exists a $\mathbf{T}$-transversal Hamilton path. Moreover, if $m=|V|$ and at least $m-1$ of the tournaments $T_1,\ldots,T_m$ are assumed to be strongly connected, then there is a $\mathbf{T}$-transversal Hamilton cycle. In our proof, we utilize a novel way of partitioning tournaments which we dub $\mathbf{H}$-partition.

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arXiv2024

Twin-width of subdivisions of multigraphs

For each $d\leq3$, we construct a finite set $F_d$ of multigraphs such that for each graph $H$ of girth at least $5$ obtained from a multigraph $G$ by subdividing each edge at least two times, $H$ has twin-width at most $d$ if and only if $G$ has no minor in $F_d$. This answers a question of Bergé, Bonnet, and Déprés asking for the structure of graphs $G$ such that each long subdivision of $G$ has twin-width $4$. As a corollary, we show that the $7\times7$ grid has twin-width $4$, which answers a question of Schidler and Szeider.

math.CO↗

arXiv2024

Twin-width of random graphs

We investigate the twin-width of the Erdős-Rényi random graph $G(n,p)$. We unveil a surprising behavior of this parameter by showing the existence of a constant $p^*\approx 0.4$ such that with high probability, when $p^*\le p\le 1-p^*$, the twin-width is asymptotically $2p(1-p)n$, whereas, when $0 p>1-p^*$, the twin-width is significantly higher than $2p(1-p)n$. In addition, we show that the twin-width of $G(n,1/2)$ is concentrated around $n/2 - \sqrt{3n \log n}/2$ within an interval of length $o(\sqrt{n\log n})$. For the sparse random graph, we show that with high probability, the twin-width of $G(n,p)$ is $Θ(n\sqrt{p})$ when $(726\ln n)/n\leq p\leq1/2$.

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arXiv2024

Edge-disjoint cycles with the same vertex set

In 1975, Erdős asked for the maximum number of edges that an $n$-vertex graph can have if it does not contain two edge-disjoint cycles on the same vertex set. It is known that Turán-type results can be used to prove an upper bound of $n^{3/2+o(1)}$. However, this approach cannot give an upper bound better than $Ω(n^{3/2})$. We show that, for any $k\geq 2$, every $n$-vertex graph with at least $n \cdot \mathrm{polylog}(n)$ edges contains $k$ pairwise edge-disjoint cycles with the same vertex set, resolving this old problem in a strong form up to a polylogarithmic factor. The well-known construction of Pyber, Rödl and Szemerédi of graphs without $4$-regular subgraphs shows that there are $n$-vertex graphs with $Ω(n\log \log n)$ edges which do not contain two cycles with the same vertex set, so the polylogarithmic term in our result cannot be completely removed. Our proof combines a variety of techniques including sublinear expanders, absorption and a novel tool for regularisation, which is of independent interest. Among other applications, this tool can be used to regularise an expander while still preserving certain key expansion properties.

math.CO↗

arXiv2024

Rainbow saturation for complete graphs

We call an edge-colored graph rainbow if all of its edges receive distinct colors. An edge-colored graph $Γ$ is called $H$-rainbow saturated if $Γ$ does not contain a rainbow copy of $H$ and adding an edge of any color to $Γ$ creates a rainbow copy of $H$. The rainbow saturation number $sat(n,{R}(H))$ is the minimum number of edges in an $n$-vertex $H$-rainbow saturated graph. Girão, Lewis, and Popielarz conjectured that $sat(n,{R}(K_r))=2(r-2)n+O(1)$ for fixed $r\geq 3$. Disproving this conjecture, we establish that for every $r\geq 3$, there exists a constant $α_r$ such that $$r + Ω\left(r^{1/3}\right) \le α_r \le r + r^{1/2} \qquad \text{and} \qquad sat(n,{R}(K_r)) = α_r n + O(1).$$ Recently, Behague, Johnston, Letzter, Morrison, and Ogden independently gave a slightly weaker upper bound which was sufficient to disprove the conjecture. They also introduced the weak rainbow saturation number, and asked whether this is equal to the rainbow saturation number of $K_r$, since the standard weak saturation number of complete graphs equals the standard saturation number. Surprisingly, our lower bound separates the rainbow saturation number from the weak rainbow saturation number, answering this question in the negative. The existence of the constant $α_r$ resolves another of their questions in the affirmative for complete graphs. Furthermore, we show that the conjecture of Girão, Lewis, and Popielarz is true if we have an additional assumption that the edge-colored $K_r$-rainbow saturated graph must be rainbow. As an ingredient of the proof, we study graphs which are $K_r$-saturated with respect to the operation of deleting one edge and adding two edges.

math.CO↗

arXiv2024

Exact results on generalized Erdős-Gallai problems

Generalized Turán problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem is maximizing the number of cliques of size $t$ in a graph of a fixed order that does not contain any path (or cycle) of length at least a given number. Both of the path-free and cycle-free extremal problems were recently considered and asymptotically solved by Luo. We fully resolve these problems by characterizing all possible extremal graphs. We further extend these results by solving the edge-variant of these problems where the number of edges is fixed instead of the number of vertices. We similarly obtain exact characterization of the extremal graphs for these edge variants.

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arXiv2024

Almost spanning distance trees in subsets of finite vector spaces

For $d\ge 2$ and an odd prime power $q$, consider the vector space $\mathbb{F}_q^d$ over the finite field $\mathbb{F}_q$, where the distance between two points $(x_1,\ldots,x_d)$ and $(y_1,\ldots,y_d)$ is defined as $\sum_{i=1}^d (x_i-y_i)^2$. A distance graph is a graph associated with a non-zero distance to each of its edges. We show that large subsets of vector spaces over finite fields contain every nearly spanning distance tree with bounded degree in each distance. This quantitatively improves results by Bennett, Chapman, Covert, Hart, Iosevich, and Pakianathan on finding distance paths, and results by Pham, Senger, Tait, and Thu on finding distance trees. A key ingredient in proving our main result is to obtain a colorful generalization of a classical result of Haxell about finding nearly spanning bounded-degree trees in an expander.

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arXiv2023

Extremal bipartite independence number and balanced coloring

In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, we show that every sufficiently large bipartite graph with average degree $D$ and with $n$ vertices on each side has a balanced independent set containing $(1-ε) \frac{\log D}{D} n$ vertices from each side for small $ε> 0$. Secondly, we prove that the vertex set of every sufficiently large balanced bipartite graph with maximum degree at most $Δ$ can be partitioned into $(1+ε)\fracΔ{\log Δ}$ balanced independent sets. Both of these results are algorithmic and best possible up to a factor of 2, which might be hard to improve as evidenced by the phenomenon known as `algorithmic barrier' in the literature. The first result improves a recent theorem of Axenovich, Sereni, Snyder, and Weber in a slightly more general setting. The second result improves a theorem of Feige and Kogan about coloring balanced bipartite graphs.

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