arXiv subjects

Thomas Shermer

Publications and source records attributed to Thomas Shermer.

4 recordsLinked to original sources

arXiv2026

Polychromatic 2-colorings with Bounded Discrepancy for Triangulations

A polychromatic $2$-coloring of a triangulation is a $2$-coloring of the vertices such that no face is monochromatic. The discrepancy of a coloring is the maximum difference between the sizes of the color classes. Asayama and Matsumoto (Graphs and Combinatorics, 2022) proved that every triangulation admits a polychromatic $2$-coloring with discrepancy at most $\tfrac{5n-16}{9}$, and that there exists a class of triangulations for which every polychromatic $2$-coloring has discrepancy at least $\tfrac{n}{3} - 2$, where $n$ is the number of vertices. We improve the upper bound, showing that every triangulation admits a polychromatic $2$-coloring with discrepancy at most $\tfrac{3n-16}{7}$ and such a $2$-coloring can be computed in quadratic time. We also show a discrepancy of at most $n-\tfrac{4M}{3}$ for triangulations with a matching of size $M$. This implies, for example, that Delaunay triangulations admit a discrepancy of at most $\tfrac{n}{3}$. We provide a linear-time algorithm to compute a $2$-coloring whose discrepancy is at most $\tfrac{5n-24}{7}$. One of our results shows that any proper four coloring with the largest color class of size $\frac{n}{2}$ would imply a $2$-coloring with discrepancy at most $\frac{n}{3}$. The existence of such a proper coloring has been recently confirmed by Kawarabayashi, Yoneda, and Yoneda (arXiv 2026). Therefore the two results together confirm the discrepancy of at most $\frac{n}{3}$ for triangulations.

cs.CG↗

arXiv2025

Contiguous Boundary Guarding

We study the problem of guarding the boundary of a simple polygon with a minimum number of guards such that each guard covers a contiguous portion of the boundary. First, we present a simple greedy algorithm for this problem that returns a guard set of size at most OPT + 1, where OPT is the number of guards in an optimal solution. Then, we present a polynomial-time exact algorithm. While the algorithm is not complicated, its correctness proof is rather involved. This result is interesting in the sense that guarding problems are typically NP-hard and, in particular, it is NP-hard to minimize the number of guards to see the boundary of a simple polygon, without the contiguous boundary guarding constraint. From the combinatorial point of view, we show that any $n$-vertex polygon can be guarded by at most $\lfloor \frac{n-2}{2}\rfloor$ guards. This bound is tight because there are polygons that require this many guards.

cs.CG↗

arXiv2022

Pursuit-Evasion in Graphs: Zombies, Lazy Zombies and a Survivor

We study zombies and survivor, a variant of the game of cops and robber on graphs. In this variant, the single survivor plays the role of the robber and attempts to escape from the zombies that play the role of the cops. The zombies are restricted, on their turn, to always follow an edge of a shortest path towards the survivor. Let $z(G)$ be the smallest number of zombies required to catch the survivor on a graph $G$ with $n$ vertices. We show that there exist outerplanar graphs and visibility graphs of simple polygons such that $z(G) = Θ(n)$. We also show that there exist maximum-degree-$3$ outerplanar graphs such that $z(G) = Ω\left(n/\log(n)\right)$. Let $z_L(G)$ be the smallest number of lazy zombies (zombies that can stay still on their turn) required to catch the survivor on a graph $G$. We establish that lazy zombies are more powerful than normal zombies but less powerful than cops. We prove that $z_L(G) = 2$ for connected outerplanar graphs. We show that $z_L(G)\leq k$ for connected graphs with treedepth $k$. This result implies that $z_L(G)$ is at most $(k+1)\log n$ for connected graphs with treewidth $k$, $O(\sqrt{n})$ for connected planar graphs, $O(\sqrt{gn})$ for connected graphs with genus $g$ and $O(h\sqrt{hn})$ for connected graphs with any excluded $h$-vertex minor. Our results on lazy zombies still hold when an adversary chooses the initial positions of the zombies.

math.CO↗

arXiv2018

Gathering by Repulsion

We consider a repulsion actuator located in an $n$-sided convex environment full of point particles. When the actuator is activated, all the particles move away from the actuator. We study the problem of gathering all the particles to a point. We give an $O(n^2)$ time algorithm to compute all the actuator locations that gather the particles to one point with one activation, and an $O(n)$ time algorithm to find a single such actuator location if one exists. We then provide an $O(n)$ time algorithm to place the optimal number of actuators whose sequential activation results in the gathering of the particles when such a placement exists.

cs.CG↗