arXiv subjects

Aashirwad Mohapatra

Publications and source records attributed to Aashirwad Mohapatra.

6 recordsLinked to original sources

arXiv2026

Solving Vertex Integrity Faster than $2^n$

Vertex Integrity is a classical graph parameter measuring the vulnerability of a network to vertex failures. We settle two fundamental questions concerning its exact exponential complexity. First, we prove that, unless the Exponential Time Hypothesis fails, Vertex Integrity has no $2^{o(n)}n^{O(1)}$-time algorithm. Secondly, we give a deterministic exact algorithm running in $O(1.9602^n)$ time and space, improving the straightforward $O(2^n)$ bound.

cs.DS↗

arXiv2026

Strong Edge Colouring of Disk Graphs: A 6-Approximation and an Improved Unit-Disk Bound

A strong edge colouring of a graph $G$ is an edge colouring in which every colour class is an induced matching. The minimum number of colours is the strong chromatic index $χ'_s(G)$. If each edge $e$ is assigned a list $L'(e)$ and its colour must belong to $L'(e)$, the corresponding parameter is the strong list chromatic index $χ'_{s,\ell}(G)$. From the definitions, $χ'_s(G)\leχ'_{s,\ell}(G)$. Barrett et al. gave an $8$-approximation for strong edge colouring on unit disk graphs and Grelier et al. improved the approximation factor to $6$. Our first result extends this factor-$6$ guarantee from unit disk graphs to the strictly larger class of disk graphs. In another direction, Erdős and Nešetřil conjectured that the strong chromatic index of a graph of maximum degree $Δ$ is asymptotically at most $1.25Δ^2$. The best published general asymptotic upper bound has leading coefficient $1.772$, due to Hurley et al. For unit disk graphs, Dębski et al. proved that $χ'_s(G) \leq 1.625 Δ^2$. Our second result improves this leading coefficient to $225/142 \approx 1.5845$. In fact, the proof establishes a stronger bound $χ'_{s,\ell}(G)\le\frac{225}{142} Δ^2+O(Δ)$ for unit disk graphs.

math.CO↗

arXiv2026

Large Planar Point Sets Contain 4 Collinear Points or Almost 7-Cliques, and Related Results

We prove that every sufficiently large finite planar point set contains either four collinear points or seven points with at most one non-visible pair. More generally, we show that for every fixed graph $H$ with chromatic number at most five, or with chromatic number six and a color-critical edge, the visibility graph of every sufficiently large finite planar point set with no four collinear points contains a copy of $H$. These results extend the recent breakthrough of Bonnet (2026), guaranteeing six pairwise visible points, and come within one visibility edge of the next open case of the big-line-big-clique conjecture.

math.CO↗

arXiv2026

Almost Empty Monochromatic Triangles With Many Colors

Given integers $c\geq 2$ and $s\geq 0$, let $\mathsf{M}_3(c,s)$ denote the least integer such that every set of at least $\mathsf{M}_3(c,s)$ points in the plane, no three on a line, colored with $c$ colors, contains a monochromatic triangle with at most $s$ interior points. Further, let $λ_3(c)$ be the least integer such that $\mathsf{M}_3(c,λ_3(c))<\infty$. \citet{colorempty} proved that, for every $c\geq 2$, $$\left\lfloor\frac{c-1}{2}\right\rfloor \leq λ_3(c)\leq c-2.$$ Later, \citet{cravioto2019almost} improved the upper bound to $c-3$, for $c\geq 4$. In this paper, we refine their argument to obtain the following asymptotic improvement: $$λ_3(c) \leq c-\sqrt{c\log c}+o (\sqrt{c\log c} ),$$ for all sufficiently large $c$. We also show that every $c$-coloring of a sufficiently large Horton set contains a monochromatic triangle with at most $\lfloor \frac{c-1}{2} \rfloor$ interior points. This shows that the aforementioned lower bound on $λ_3(c)$ is sharp within the class of Horton sets. We conclude with a conjecture on the large-color asymptotics of $λ_3(c)$.

math.CO↗

arXiv2026

On the Number of Almost Empty Monochromatic Triangles

In this paper, we consider the problem of counting almost empty monochromatic triangles in colored planar point sets, that is, triangles whose vertices are all assigned the same color and that contain only a few interior points. Specifically, we show that any $c$-coloring of a set of $n$ points in the plane in general position (that is, no three on a line) contains $Ω(n^2)$ monochromatic triangles with at most $c-1$ interior points and $Ω(n^{\frac{4}{3}})$ monochromatic triangles with at most $c-2$ interior points, for any fixed $c \geq 2$. The latter, in particular, generalizes the result of Pach and Tóth (2013) on the number of monochromatic empty triangles in 2-colored point sets, to the setting of multiple colors and monochromatic triangles with a few interior points. We also derive the limiting value of the expected number of triangles with $s$ interior points in random point sets, for any integer $s \geq 0$. As a result, we obtain the expected number of monochromatic triangles with at most $s$ interior points in random colorings of random point sets.

math.CO↗

arXiv2026

An Improved Upper Bound for the Turán Number of the Hexagon

For a graph $F$, the Turán number $\operatorname{ex}(n,F)$ is the maximum number of edges in an $n$-vertex graph containing no copy of $F$. Determining the Turán numbers of even cycles is a central problem in extremal graph theory and remains open in general. For $C_6$, the best previous upper bound was due to Füredi, Naor, and Verstraëte [Advances in Mathematics, 2006], who proved that, for sufficiently large positive integer $n$, $$ \operatorname{ex}(n,C_6) \leq λn^{4/3}+O(n)<0.6272 n^{4/3}, $$ where $λ$ is the real root of $ 16λ^3-4λ^2+λ-3=0$. We improve this bound by showing that, for sufficiently large positive integer $n$, $$ \operatorname{ex}(n,C_6) \leq αn^{4/3}+O(n)<0.6144 n^{4/3}, $$ where $α$ is the unique real root of $ 4 α^{3} (3/2)^{1-1/(2α)} =1$ in the interval $(1/2,2/3)$.

math.CO↗