arXiv subjects

Padmini Mukkamala

Publications and source records attributed to Padmini Mukkamala.

4 recordsLinked to original sources

arXiv2025

Triangle-free subsets of the $r$-distance graph of the hypercube

Given the $r$-distance graph on the hypercube $\F_2^n$, where two vertices are adjacent if their Hamming distance is exactly $r$, we study the maximum size $T(n,r)$ of a triangle-free set of vertices. For even $r\le n/2$, we prove \[ T(n,r)=O\!\left(\frac{r2^n}{n+1}\right). \] In particular, $T(n,r)=o(2^n)$ whenever $r=o(n)$. For fixed $0<\alpha<2/3$, we also prove that if $r=\alpha n$, where $n$ ranges over integers such that $\alpha n$ is an even integer, then \[ T(n,r)\le 2^{(1-\varepsilon_\alpha)n} \] for some $\varepsilon_\alpha>0$. We also obtain lower bounds in various regimes of $r$ as a function of $n$.

math.CO↗

arXiv2011

Drawing cubic graphs with the four basic slopes

We show that every cubic graph can be drawn in the plane with straight-line edges using only the four basic slopes $\{0,π/4,π/2,3π/4\}$. We also prove that four slopes have this property if and only if we can draw $K_4$ with them.

math.CO↗

arXiv2011

Lower bounds on the obstacle number of graphs

Given a graph $G$, an {\em obstacle representation} of $G$ is a set of points in the plane representing the vertices of $G$, together with a set of connected obstacles such that two vertices of $G$ are joined by an edge if and only if the corresponding points can be connected by a segment which avoids all obstacles. The {\em obstacle number} of $G$ is the minimum number of obstacles in an obstacle representation of $G$. It is shown that there are graphs on $n$ vertices with obstacle number at least $Ω({n}/{\log n})$.

math.CO↗

arXiv2010

Almost optimal pairing strategy for Tic-Tac-Toe with numerous directions

We show that there is an $m=2n+o(n)$, such that, in the Maker-Breaker game played on $\Z^d$ where Maker needs to put at least $m$ of his marks consecutively in one of $n$ given winning directions, Breaker can force a draw using a pairing strategy. This improves the result of Kruczek and Sundberg who showed that such a pairing strategy exits if $m\ge 3n$. A simple argument shows that $m$ has to be at least $2n+1$ if Breaker is only allowed to use a pairing strategy, thus the main term of our bound is optimal.

math.CO↗