On Game Chromatic Vertex-Critical Graphs

While the game chromatic number of a forest is known to be at most 4, no simple criteria are known for determining the game chromatic number of a forest. We first state necessary and sufficient conditions for forests with game chromatic number 2 and then investigate the differences between forests with game chromatic number 3 and 4. In doing so, we present a minimal example of a forest with game chromatic number 4, criteria for determining in polynomial time the game chromatic number of a forest without vertices of degree 3, and an example of a forest with maximum degree 3 and game chromatic number 4. This gives partial progress on the open question of the computational complexity of the game chromatic number of a forest.

This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G* be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G* has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product G□G′ has game chromatic number at most k(k + m − 1). As a consequence, the Cartesian product of two forests has game chromatic number at most 10, and the Cartesian product of two planar graphs has game chromatic number at most 105. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 261–278, 2008

2-Coloring number revisited

Graph coloring is the process of assigning colors to the vertices or edges of a graph. Specifically, coloring the vertices in graph coloring can be implemented in graph coloring games. This article aims to determine the game chromatic number of the jellyfish graph Jm,n, snail graph SIn, and octopus graph On. For example, give G as a simple, connected, and undirected graph and give two players, namely A as the first player and B as the secondary player. The two players A and B color all the vertices of graph G with available colors. The game’s rules are that A must ensure that all vertices of graph G are colored, while B aims to prevent graph G from being uncolored. Players A and B take turns coloring the vertices of graph G, ensuring that the color of neighboring vertices must be different, with A taking the first turn. If all vertices have been colored, A wins the game, but A loses if some vertices remain uncolored despite available colors, A loses. The smallest value of k for which A has a winning strategy in the game with k colors is the game chromatic number, denoted as χg(G). This thesis discusses the graph coloring game of the tadpole graph Tm,n, broom graph Bn,d, jellyfish graph Jm,n, and tribune graph Tn to find the game chromatic number. The results show that player A uses a strategy to color the vertex with the highest degree in the graph, ensuring that player A always wins. Therefore, the game chromatic number of the jellyfish graph, snail graph, and octopus graph is χg (Jm,n) = 3 for m, n ≥ 1, and χg (SIn) = 3 for n ≥ 1, while χg (On) = 3 for n = 2, 3, 4; χg (On) = 4 for n ≥ 5.

Game-perfect Graphs with Clique Number 2

We propose a new coloring game on a graph, called the independence coloring game, which is played by two players with opposite goals. The result of the game is a proper coloring of vertices of a graph G, and Alice’s goal is that as few colors as possible are used during the game, while Bob wants to maximize the number of colors. The game consists of rounds, and in round i, where i = 1, 2,, … , the players are taking turns in selecting a previously unselected vertex of G and giving it color i (hence, in each round the selected vertices form an independent set). The game ends when all vertices of G are selected (and thus colored), and the total number of rounds during the game when both players are playing optimally with respect to their goals, is called the independence game chromatic number, χ ig (G), of G. In fact, four different versions of the independence game chromatic number are considered, which depend on who starts a game and who starts next rounds. We prove that the new invariants lie between the chromatic number of a graph and the maximum degree plus 1, and characterize the graphs in which each of the four versions of the game invariant equals 2. We compare the versions of the independence game chromatic number among themselves and with the classical game chromatic number. In addition, we prove that the independence game chromatic number of a tree can be arbitrarily large.

The coloring game on planar graphs with large girth, by a result on sparse cactuses

Autotopism stabilized colouring games on rook’s graphs

Let G = V , E be a graph, and two players Alice and Bob alternate turns coloring the vertices of the graph G a proper coloring where no two adjacent vertices are signed with the same color. Alice's goal is to color the set of vertices using the minimum number of colors, which is called game chromatic number and is denoted by χ g G , while Bob's goal is to prevent Alice's goal. In this paper, we investigate the game chromatic number χ g G of Generalized Petersen Graphs G P n , k for k ≥ 3 and arbitrary n , n -Crossed Prism Graph, and Jahangir Graph J n , m .

A graph G is called g-perfect if, for any induced subgraph H of G, the game chromatic number of H equals the clique number of H. A graph G is called g-col-perfect if, for any induced subgraph H of G, the game coloring number of H equals the clique number of H. In this paper we characterize the classes of g-perfect resp. g-col-perfect graphs by a set of forbidden induced subgraphs and explicitly. Moreover, we study similar notions for variants of the game chromatic number, namely B-perfect and [A,B]-perfect graphs, and for several variants of the game coloring number, and characterize the classes of these graphs.

Given a graph $G$ and an integer $k$, two players take turns coloring the vertices of $G$ one by one using $k$ colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of $G$ are colored. The game chromatic number $\chi_g(G)$ is the minimum $k$ for which the first player has a winning strategy. The paper [T. Bohman, A. M. Frieze, and B. Sudakov, Random Structures Algorithms, 32 (2008), pp. 223--235] began the analysis of the asymptotic behavior of this parameter for a random graph $G_{n,p}$. This paper provides some further analysis for graphs with constant average degree, i.e., $np=O(1)$, and for random regular graphs. We show that with high probability (w.h.p.) $c_1\chi(G_{n,p})\leq \chi_g(G_{n,p})\leq c_2\chi(G_{n,p})$ for some absolute constants $1<c_1< c_2$. We also prove that if $G_{n,3}$ denotes a random $n$-vertex cubic graph, then w.h.p. $\chi_g(G_{n,3})=4$.

The M‐degree of an edge xy in a graph is the maximum of the degrees of x and y. The M‐degree of a graph G is the minimum over M‐degrees of its edges. In order to get upper bounds on the game chromatic number, He et al showed that every planar graph G without leaves and 4‐cycles has M‐degree at most 8 and gave an example of such a graph with M‐degree 3. This yields upper bounds on the game chromatic number of C4‐free planar graphs. We determine the maximum possible M‐degrees for planar, projective‐planar and toroidal graphs without leaves and 4‐cycles. In particular, for planar and projective‐planar graphs this maximum is 7. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 80–85, 2009

Directed defective asymmetric graph coloring games

In the paper we introduce the new game--the unilateral $${\mathcal{P}}$$ -colouring game which can be used as a tool to study the r-colouring game and the (r, d)-relaxed colouring game. Let be given a graph G, an additive hereditary property $${\mathcal {P}}$$ and a set C of r colours. In the unilateral $${\mathcal {P}}$$ -colouring game similarly as in the r-colouring game, two players, Alice and Bob, colour the uncoloured vertices of the graph G, but in the unilateral $${\mathcal {P}}$$ -colouring game Bob is more powerful than Alice. Alice starts the game, the players play alternately, but Bob can miss his move. Bob can colour the vertex with an arbitrary colour from C, while Alice must colour the vertex with a colour from C in such a way that she cannot create a monochromatic minimal forbidden subgraph for the property $${\mathcal {P}}$$ . If after |V(G)| moves the graph G is coloured, then Alice wins the game, otherwise Bob wins. The $${\mathcal {P}}$$ -unilateral game chromatic number, denoted by $${\chi_{ug}^\mathcal {P}(G)}$$ , is the least number r for which Alice has a winning strategy for the unilateral $${\mathcal {P}}$$ -colouring game with r colours on G. We prove that the $${\mathcal {P}}$$ -unilateral game chromatic number is monotone and is the upper bound for the game chromatic number and the relaxed game chromatic number. We give the winning strategy for Alice to play the unilateral $${\mathcal {P}}$$ -colouring game. Moreover, for k ≥ 2 we define a class of graphs $${\mathcal {H}_k =\{G|{\rm every \;block \;of\;}G \; {\rm has \;at \;most}\; k \;{\rm vertices}\}}$$ . The class $${\mathcal {H}_k }$$ contains, e.g., forests, Husimi trees, line graphs of forests, cactus graphs. Let $${\mathcal {S}_d}$$ be the class of graphs with maximum degree at most d. We find the upper bound for the $${\mathcal {S}_2}$$ -unilateral game chromatic number for graphs from $${\mathcal {H}_3}$$ and we study the $${\mathcal {S}_d}$$ -unilateral game chromatic number for graphs from $${\mathcal {H}_4}$$ for $${d \in \{2,3\}}$$ . As the conclusion from these results we obtain the result for the d-relaxed game chromatic number: if $${G \in \mathcal {H}_k}$$ , then $${\chi_g^{(d)}(G) \leq k + 2-d}$$ , for $${k \in \{3, 4\}}$$ and $${d \in \{0, \ldots, k-1\}}$$ . This generalizes a known result for trees.