31 problems

Let $G$ be a bipartite graph of even order, with nonnegative vertex weights, and consider the graph-grabbing game in which Alice moves first and vertices are removed while the rema…

A part is a connected component of a linear clobber position. Let $\mathcal{A}$ be the set of non-empty even-length alternating-color parts, namely … In normal-play combinatorial g…

Let $W_n$ be a wheel graph and let $C(n,n)$ be the corresponding Ferris Wheel graph, for $n\geq 3$. Suppose Player X wins on $W_n$ with score $(a,b)$. Ferris Wheel score conjecture…

Let $C(n,k)$ be a loopy cycle formed from a cycle $C_n$ by adding $k$ loops to consecutive vertices, with $n\geq 3$. Loopy-cycle winner conjecture. For $k=1$ and $k=2$, Player 1 wi…

Let $G$ be a graph, and let $G'$ be obtained by adding a single loop to an existing vertex of $G$. Loop addition conjecture. Adding a single loop to an existing vertex always swaps…

Let $T$ be a tree of order $n$, and let $P_n$ denote the path on $n$ vertices. Path extremal conjecture. The game cordiality number satisfies … This conjecture proposes that paths…

Fay, Hurlbert and Tennant's stackability conjecture. The hypercube $Q_d$ is stackable for all $d\geq 1$, and, more generally, every Cartesian product of paths of arbitrary lengths…

Let $G$ be a graph of even order, and let $\mathcal{D}$ be the family of graphs shown in the paper's figure, where the cycle has an odd number of vertices and any subset of the das…

Let $P_k$ be the path graph on $k$ vertices, let $d$ be a positive integer, and let $k_1,\ldots,k_d$ be positive integers. Write the Cartesian product grid as … Call a graph stacka…

Let $Q^d$ denote the $d$-dimensional hypercube, and call a graph stackable if, starting with one cup at each vertex, all cups can be moved onto any prescribed target vertex accordi…

Let $G$ be either the diagonal square lattice or the hexagonal lattice, and let $q(G,f(t))$ denote the set of containment rates achievable when the Container has spread function…

Let $\mathbb Z^2_{\triangle}$ denote the triangular lattice, and let $q_{\mathbb Z^2_{\triangle}}$ be its containment threshold in the firefighter problem. The triangular-grid fire…

Consider a cop and a robber playing on an arbitrary graph. A winning strategy is monotone if the cop never purposefully increases the distance between the cop and the robber. Monot…

Let a cop and a robber play the game on a tree, with the cop using a strategy to win the game. The strategy CS means that the cop always moves in the direction of the robber. Cop s…

Let $G_{2,m}$ be the rectangular grid with two rows and $m$ columns, and let $Ls(G_{2,m})$ and $Rs(G_{2,m})$ denote the corresponding left and right scores. Two-row grid score conj…

Let $G_{n,m}$ be the rectangular grid with $n$ rows and $m$ columns, with alternated black and white vertices and a black vertex in the top-left corner. Let $Ls(G_{n,m})$ and…

A segment of $n$ vertices starting with a black vertex is denoted by $S_n$, and a segment starting with a white vertex by $S_{-n}$. Let $Ls(S_n)$ and $Rs(S_n)$ denote the left and…

Let $G$ be a fixed graph with degeneracy $d$, and let $\tau_{\mathcal{P}_G}$ denote the threshold exponent for forcing a graph containing a copy of $G$ in the semi-random graph pro…

Let $G$ be a graph of order $n$, and let $\delta(G)$ denote its minimum degree. In the Minimizer-start competition-enclaveless game, Minimizer starts, and $\Psi_g^-(G)$ is the numb…

Let $G$ be a graph of order $n$ with no isolated vertices. In the competition-enclaveless game, Maximizer starts, and $\Psi_g^+(G)$ is the number of vertices chosen when both playe…

Let $S_{m\times n}$ denote the $m\times n$ square grid, and let the $(p,p)$-crossing game be the Maker-Breaker game in which Maker and Breaker claim equal numbers of edges per roun…

Let $S_{m\times n}$ denote the $m\times n$ square grid, and let the $(p,q)$-crossing game be the Maker-Breaker game on this grid in which Maker claims $p$ edges for every $q$ edges…

A subdivided star is a graph consisting of three rays joined at a common central vertex, with the size of a ray equal to its number of edges. In Arc-Kayles, a move removes an edge,…

Let $KAYLES_n$ denote the Kayles graph game on a path of edge-length $n$, and let $\boxplus$ denote the selective compound operation used in the paper; write $*1$ for the one-point…

Let $STARKAYLES_{k,n}$ be the graph game in which the underlying graph is obtained by starting with a star graph on $k$ vertices and extending one branch to a path of edge-length…