Candy-passing Games on General Graphs, I

Candy-passing Games on General Graphs, II

We give a new proof that any candy-passing game on a graph G with at least 4|E(G)|-|V(G)| candies stabilizes. (This result was first proven in arXiv:0807.4450.) Unlike the prior literature on candy-passing games, we use methods from the general theory of chip-firing games which allow us to obtain a polynomial bound on the number of rounds before stabilization.

The Candy-Passing Game for c\geq3n-2

We determine the behavior of Tanton's candy-passing game for all distributions of at least 3n-2 candies, where n is the number of students. Specifically, we show that the configuration of candy in such a game eventually becomes fixed.

$\mathcal{P}$ Play in Candy Nim

Candy Nim is a variant of Nim in which both players aim to take the last candy in a game of Nim, with the added simultaneous secondary goal of taking as many candies as possible. We give bounds on the number of candies the first and second players obtain in 3-pile $\mathcal{P}$ positions as well as strategies that are provably optimal for some families of such games. We also show how to construct a game with $N$ candies such that the loser takes the largest possible number of candies and bound the number of candies the winner can take in an arbitrary $\mathcal{P}$ position with $N$ total candi…

Obstacles, Slopes, and Tic-Tac-Toe: An excursion in discrete geometry and combinatorial game theory

A drawing of a graph is said to be a {\em straight-line drawing} if the vertices of $G$ are represented by distinct points in the plane and every edge is represented by a straight-line segment connecting the corresponding pair of vertices and not passing through any other vertex of $G$. The minimum number of slopes in a straight-line drawing of $G$ is called the slope number of $G$. We show that every cubic graph can be drawn in the plane with straight-line edges using only the four basic slopes $\{0,π/4,π/2,-π/4\}$. We also prove that four slopes have this property if and only if we can draw …

Bounding Game Temperature Using Confusion Intervals

For combinatorial games, temperature is a measure of the volatility, that is, by how much the advantage can change. Typically, the temperature has been measured for individual positions within specific games. In this paper, we give the first general upper bounds on the temperature for any class of games. For a position $G$, the closure of the set of numbers $\{g\}$ such that $G-g$ is a first player win, is called the confusion interval of $G$. Let $\ell(G)$ be the length of this interval. Our first main result is: For a class of games $\mathscr{S}$, if there are constants $J$ and $K$ such that…

关于弦图的警察数

On the Cop Number of String Graphs

本文针对弦图警察数上界这一开放问题，提出新颖的守卫技术，将上界从15改进至13，并将该技术推广到完全活跃警察与强盗变体及boxicity 2图，同时得到围长至少为5的弦图色数至多为14。