Consider the following two-person game on a graph \( G \). The two players start with two color choices only, taking turns coloring any uncolored vertex with the restriction that any coloring must be a proper coloring. A third (or fourth, etc.) color can only be used when forced to maintain a proper coloring. One player, the minimizer, is trying to force the smallest number of colors possible. The other player, the maximizer, is trying to force the largest number of colors possible. This game proper chromatic number, denoted \( \chi_{(E,g)}(G) \), is the minimum number of colors used when both players play optimally.

The advantage of the game proper chromatic number is that it is comparable to other published game chromatic variants, particularly the game chromatic number II and the game Grundy number.

This paper also considers extensions of the game proper chromatic number through generalized regions of the graph. Let \( R = \{R_1, R_2, \ldots, R_t\} \) such that \( \bigcup R_i = V(G) \). It is convenient to think of these \( R_i \)’s as regions of interest in graph \( G \). In particular, extensions to closed neighborhoods and open neighborhoods maintaining the restriction that all colorings must be “proper” in the sense that no \( R_i \) is monochromatic are considered for some natural classes of graphs.

The minimum number of colors necessary provided each player plays optimally, following the rules established for the game proper chromatic number, is denoted \( \chi_{(N[v],g)}(G) \) and \( \chi_{(N(v),g)}(G) \) for the game closed neighborhood proper chromatic number and the game open neighborhood chromatic number, respectively.

A conjecture by Albertson states that if \( \chi(G) \geq n \), then \( cr(G) \geq cr(K_n) \), where \( \chi(G) \) is the chromatic number of \( G \) and \( cr(G) \) is the crossing number of \( G \). This conjecture is true for \( n \leq 16 \), but it remains open for \( n \geq 17 \).

In this paper, we consider the statements corresponding to this conjecture where the crossing number of \( G \) is replaced with:
– the genus \( \gamma(G) \) (the minimum genus of the orientable surface on which \( G \) is embeddable),
– the skewness \( \mu(G) \) (the minimum number of edges whose removal makes \( G \) planar), and
– the thickness \( \theta(G) \) (the minimum number of planar subgraphs of \( G \) whose union is \( G \)).

In 2017, Hedetniemi asked the question: “For which graphs \( G \) does the indexed family \( \{N_G(v) \mid v \in V(G)\} \) of open neighborhoods have a system of distinct representatives?” In [1], we answered that question. Now, we move on to other special set families in graphs and examine whether they do or do not have a system of distinct representatives.

We give necessary and sufficient conditions for two matroids on the same ground set to be the upper and lower matroids of a \( \Delta \)-matroid.

Let \( D \) be a digraph on \( n \) vertices. A cycle \( C \) in \( D \) is said to be 1-extendable if there exists a cycle \( C’ \) in \( D \) such that the vertex set of \( C’ \) contains the vertex set of \( C \) and \( C’ \) contains exactly one additional vertex. A digraph is 1-cycle-extendable if every non-Hamiltonian cycle is 1-extendable.

A cycle \( C \) in \( D \) is said to be 2-extendable if there exists a cycle \( C’ \) in \( D \) such that the vertex set of \( C’ \) contains the vertex set of \( C \) and \( C’ \) contains exactly two additional vertices. A digraph is 2-cycle-extendable if every cycle on at most \( n-2 \) vertices is 2-extendable.

A digraph is 1,2-cycle-extendable if every non-Hamiltonian cycle is either 1-extendable or 2-extendable. It has been previously shown that not all strong tournaments (orientations of a complete undirected graph) are 1-extendable, but are 2-extendable. The structure of all non 1-extendable tournaments is shown as a type of block Kronecker product of 1-extendable subtournaments.

For a toroidal graph \( G = (V, E) \) embedded in the torus, let \( \mathcal{F}(G) \) denote the set of faces of \( G \). Then, \( G \) is called a \( C_{n} \)-face-magic torus graph if there exists a bijection \( f: V(G) \rightarrow \{1, 2, \ldots, |V(G)|\} \) such that for any \( F \in \mathcal{F}(G) \) with \( F \cong C_{n} \), the sum of all the vertex labelings along \( C_{n} \) is a constant \( S \).

Let \( x_{v} = f(v) \) for all \( v \in V(G) \). We call \( \{x_{v} : v \in V(G)\} \) a \( C_{n} \)-face magic torus labeling on \( G \).

We say that a \( C_{4} \)-face-magic torus labeling \( \{x_{i,j} \} \) on \( C_{2n} \times C_{2n} \) is antipodal balanced if \( x_{i,j} + x_{i+n,j+n} = \frac{1}{2}S \) for all \( (i, j) \in V(C_{2n} \times C_{2n}) \).

We determine all antipodal balanced \( C_{4} \)-face-magic torus labelings on \( C_{4} \times C_{4} \) up to symmetries on a torus.

Steinhaus graphs are a small (there are \( 2^{n-1} \) of them on \( n \) vertices) but interesting family of graphs. They have been studied for over forty years, and it has been shown that almost all graphs have certain properties if and only if almost all Steinhaus graphs have these properties.

In this paper, we find and count all the complements of Steinhaus graphs that are claw-free.

Edge-Nim is a combinatorial game played on finite regular graphs with positive, integrally weighted edges. Two players alternately move from an initialized vertex to an adjacent vertex, decreasing the weight of the incident edge to a strictly non-negative integer as they travel across it. The game ends when no incident edge has a nonzero weight and a player is unable to move, in which case that player loses.

We characterize the winner of Edge-Nim on the complete bipartite graphs \( K_{2,n} \) for all positive integers \( n \), giving the solution and complete strategy for the player able to win.

Given a graph \( G \), we are interested in finding disjoint paths for a given set of distinct pairs of vertices. In 2017, we formally defined a new parameter, the pansophy of \( G \), in the context of the disjoint path problem.

In this paper, we investigate the pansophy of two classes of graphs that contain a vertex that we define as the superuser. The superuser of a graph is a vertex that is adjacent to every other vertex. We close with future research directions.

A grid on a cell of a game board attacks all neighboring cells. The domination number counts the minimum number of grids such that each cell of a board is occupied or attacked by a grid.

For square boards (chess boards), the domination number has been determined in a series of papers. Here, we start to consider grids on hexagon boards \( B_n \) as parts of the Euclidean tessellation by congruent regular hexagons, where \( B_1 \) is one hexagon, \( B_2 \) consists of the three hexagons around one vertex, and \( B_n \) for \( n \geq 3 \) consists of \( B_{n-2} \) together with all hexagons having at least one hexagon in common with \( B_{n-2} \).

An upper bound is presented for the grid domination number, and exact values are determined by computer for small \( n \).