Graph coloring game

The graph coloring game is a mathematical game related to graph theory. Coloring game problems arose as game-theoretic versions of well-known graph coloring problems. In a coloring game, two players use a given set of colors to construct a coloring of a graph, following specific rules depending on the game we consider. One player tries to successfully complete the coloring of the graph, when the other one tries to prevent him from achieving it.

Vertex coloring game

The vertex coloring game was introduced in 1981 by Brams^[1] and rediscovered ten years after by Bodlaender.^[2] Its rules are as follows:

1. Alice and Bob color the vertices of a graph G with a set k of colors.
2. Alice and Bob take turns, coloring properly an uncolored vertex (in the standard version, Alice begins).
3. If a vertex v is impossible to color properly (for any color, v has a neighbor colored with it), then Bob wins.
4. If the graph is completely colored, then Alice wins.

The game chromatic number of a graph ${\displaystyle G}$, denoted by ${\displaystyle \chi _{g}(G)}$, is the minimum number of colors needed for Alice to win the vertex coloring game on ${\displaystyle G}$. Trivially, for every graph ${\displaystyle G}$, we have ${\displaystyle \chi (G)\leq \chi _{g}(G)\leq \Delta (G)+1}$, where ${\displaystyle \chi (G)}$ is the chromatic number of ${\displaystyle G}$ and ${\displaystyle \Delta (G)}$ its maximum degree.^[3]

In the 1991 Bodlaender's paper,^[4] the computational complexity was left as "an interesting open problem". Only in 2020 it was proved that the game is PSPACE-Complete.^[5]

Relation with other notions

Acyclic coloring. Every graph ${\displaystyle G}$ with acyclic chromatic number ${\displaystyle k}$ has ${\displaystyle \chi _{g}(G)\leq k(k+1)}$.^[6]

Marking game. For every graph ${\displaystyle G}$, ${\displaystyle \chi _{g}(G)\leq col_{g}(G)}$, where ${\displaystyle col_{g}(G)}$ is the game coloring number of ${\displaystyle G}$. Almost every known upper bound for the game chromatic number of graphs are obtained from bounds on the game coloring number.

Cycle-restrictions on edges. If every edge of a graph ${\displaystyle G}$ belongs to at most ${\displaystyle c}$ cycles, then ${\displaystyle \chi _{g}(G)\leq 4+c}$.^[7]

Graph Classes

For a class ${\displaystyle {\mathcal {C}}}$ of graphs, we denote by ${\displaystyle \chi _{g}({\mathcal {C}})}$ the smallest integer ${\displaystyle k}$ such that every graph ${\displaystyle G}$ of ${\displaystyle {\mathcal {C}}}$ has ${\displaystyle \chi _{g}(G)\leq k}$. In other words, ${\displaystyle \chi _{g}({\mathcal {C}})}$ is the exact upper bound for the game chromatic number of graphs in this class. This value is known for several standard graph classes, and bounded for some others:

• Forests: ${\displaystyle \chi _{g}({\mathcal {F}})=4}$.^[8] Simple criteria are known to determine the game chromatic number of a forest without vertex of degree 3.^[9] It seems difficult to determine the game chromatic number of forests with vertices of degree 3, even for forests with maximum degree 3.
• Cactuses: ${\displaystyle \chi _{g}({\mathcal {C}})=5}$.^[10]
• Outerplanar graphs: ${\displaystyle 6\leq \chi _{g}({\mathcal {O}})\leq 7}$.^[11]
• Planar graphs: ${\displaystyle 7\leq \chi _{g}({\mathcal {P}})\leq 17}$.^[12]
• Planar graphs of given girth: ${\displaystyle \chi _{g}({\mathcal {P}}_{4})\leq 13}$,^[13] ${\displaystyle \chi _{g}({\mathcal {P}}_{5})\leq 8}$, ${\displaystyle \chi _{g}({\mathcal {P}}_{6})\leq 6}$, ${\displaystyle \chi _{g}({\mathcal {P}}_{8})\leq 5}$.^[14]
• Toroidal grids: ${\displaystyle \chi _{g}({{\mathcal {T}}G})=5}$.^[15]
• Partial k-trees: ${\displaystyle \chi _{g}({\mathcal {T}}_{k})\leq 3k+2}$.^[16]
• Interval graphs: ${\displaystyle 2\omega \leq \chi _{g}({\mathcal {I}})\leq 3\omega -2}$, where ${\displaystyle \omega }$ is for a graph the size of its largest clique.^[17]

Cartesian products. The game chromatic number of the cartesian product ${\displaystyle G\square H}$ is not bounded by a function of ${\displaystyle \chi _{g}(G)}$ and ${\displaystyle \chi _{g}(H)}$. In particular, the game chromatic number of any complete bipartite graph ${\displaystyle K_{n,n}}$ is equal to 3, but there is no upper bound for ${\displaystyle \chi _{g}(K_{n,n}\square K_{m,m})}$ for arbitrary ${\displaystyle n,m}$.^[18] On the other hand, the game chromatic number of ${\displaystyle G\square H}$ is bounded above by a function of ${\displaystyle {\textrm {col}}_{g}(G)}$ and ${\displaystyle {\textrm {col}}_{g}(H)}$. In particular, if ${\displaystyle {\textrm {col}}_{g}(G)}$ and ${\displaystyle {\textrm {col}}_{g}(H)}$ are both at most ${\displaystyle t}$, then ${\displaystyle \chi _{g}(G\square H)\leq t^{5}-t^{3}+t^{2}}$.^[19]

• For a single edge we have:^[18]

${\displaystyle {\begin{aligned}\chi _{g}(K_{2}\square P_{k})&={\begin{cases}2&k=1\\3&k=2,3\\4&k\geq 4\end{cases}}\\\chi _{g}(K_{2}\square C_{k})&=4&&k\geq 3\\\chi _{g}(K_{2}\square K_{k})&=k+1\end{aligned}}}$

• For stars we have:^[20]

${\displaystyle {\begin{aligned}\chi _{g}(S_{m}\square P_{k})&={\begin{cases}2&k=1\\3&k=2\\4&k\geq 3\end{cases}}\\\chi _{g}(S_{m}\square C_{k})&=4&&k\geq 3\end{aligned}}}$

• Trees: ${\displaystyle \chi _{g}(T_{1}\square T_{2})\leq 12.}$
• Wheels: ${\displaystyle \chi _{g}(P_{2}\square W_{n})=5}$ if ${\displaystyle n\geq 9.}$^[20]
• Complete bipartite graphs: ${\displaystyle \chi _{g}(P_{2}\square K_{m,n})=5}$ if ${\displaystyle m,n\geq 5.}$^[20]

Open problems

These questions are still open to this date.

Edge coloring game

The edge coloring game, introduced by Lam, Shiu and Zu,^[22] is similar to the vertex coloring game, except Alice and Bob construct a proper edge coloring instead of a proper vertex coloring. Its rules are as follows:

1. Alice and Bob are coloring the edges a graph G with a set k of colors.
2. Alice and Bob are taking turns, coloring properly an uncolored edge (in the standard version, Alice begins).
3. If an edge e is impossible to color properly (for any color, e is adjacent to an edge colored with it), then Bob wins.
4. If the graph is completely edge-colored, then Alice wins.

Although this game can be considered as a particular case of the vertex coloring game on line graphs, it is mainly considered in the scientific literature as a distinct game. The game chromatic index of a graph ${\displaystyle G}$, denoted by ${\displaystyle \chi '_{g}(G)}$, is the minimum number of colors needed for Alice to win this game on ${\displaystyle G}$.

General case

For every graph G, ${\displaystyle \chi '(G)\leq \chi '_{g}(G)\leq 2\Delta (G)-1}$. There are graphs reaching these bounds but all the graphs we know reaching this upper bound have small maximum degree.^[22] There exists graphs with ${\displaystyle \chi '_{g}(G)>1.008\Delta (G)}$ for arbitrary large values of ${\displaystyle \Delta (G)}$.^[23]

Conjecture. There is an ${\displaystyle \epsilon >0}$ such that, for any arbitrary graph ${\displaystyle G}$, we have ${\displaystyle \chi '_{g}(G)\leq (2-\epsilon )\Delta (G)}$.
This conjecture is true when ${\displaystyle \Delta (G)}$ is large enough compared to the number of vertices in ${\displaystyle G}$.^[23]

• Arboricity. Let ${\displaystyle a(G)}$ be the arboricity of a graph ${\displaystyle G}$. Every graph ${\displaystyle G}$ with maximum degree ${\displaystyle \Delta (G)}$ has ${\displaystyle \chi '_{g}(G)\leq \Delta (G)+3a(G)-1}$.^[24]

Graph Classes

For a class ${\displaystyle {\mathcal {C}}}$ of graphs, we denote by ${\displaystyle \chi '_{g}({\mathcal {C}})}$ the smallest integer ${\displaystyle k}$ such that every graph ${\displaystyle G}$ of ${\displaystyle {\mathcal {C}}}$ has ${\displaystyle \chi '_{g}(G)\leq k}$. In other words, ${\displaystyle \chi '_{g}({\mathcal {C}})}$ is the exact upper bound for the game chromatic index of graphs in this class. This value is known for several standard graph classes, and bounded for some others:

• Wheels: ${\displaystyle \chi '_{g}(W_{3})=5}$ and ${\displaystyle \chi '_{g}(W_{n})=n+1}$ when ${\displaystyle n\geq 4}$.^[22]
• Forests : ${\displaystyle \chi '_{g}({\mathcal {F}}_{\Delta })\leq \Delta +1}$ when ${\displaystyle \Delta \neq 4}$, and ${\displaystyle 5\leq \chi '_{g}({\mathcal {F}}_{4})\leq 6}$.^[25]
  Moreover, if every tree of a forest ${\displaystyle F}$ of ${\displaystyle {\mathcal {F}}_{4}}$ is obtained by subdivision from a caterpillar tree or contains no two adjacent vertices with degree 4, then ${\displaystyle \chi '_{g}(F)\leq 5}$.^[26]

Open Problems

Upper bound. Is there a constant ${\displaystyle c\geq 2}$ such that ${\displaystyle \chi '_{g}(G)\leq \Delta (G)+c}$ for each graph ${\displaystyle G}$ ? If it is true, is ${\displaystyle c=2}$ enough ?^[22]

Conjecture on large minimum degrees. There are a ${\displaystyle \epsilon >0}$ and an integer ${\displaystyle d_{0}}$ such that any graph ${\displaystyle G}$ with ${\displaystyle \delta (G)\geq d_{0}}$ satisfies ${\displaystyle \chi '_{g}(G)\geq (1+\epsilon )\delta (G)}$. ^[23]

Incidence coloring game

The incidence coloring game is a graph coloring game, introduced by Andres,^[27] and similar to the vertex coloring game, except Alice and Bob construct a proper incidence coloring instead of a proper vertex coloring. Its rules are as follows:

1. Alice and Bob are coloring the incidences of a graph G with a set k of colors.
2. Alice and Bob are taking turns, coloring properly an uncolored incidence (in the standard version, Alice begins).
3. If an incidence i is impossible to color properly (for any color, i is adjacent to an incident colored with it), then Bob wins.
4. If all the incidences are properly colored, then Alice wins.

The incidence game chromatic number of a graph ${\displaystyle G}$, denoted by ${\displaystyle i_{g}(G)}$, is the minimum number of colors needed for Alice to win this game on ${\displaystyle G}$.

For every graph ${\displaystyle G}$ with maximum degree ${\displaystyle \Delta }$, we have ${\displaystyle {\frac {3\Delta -1}{2}}<i_{g}(G)<3\Delta -1}$.^[27]

Relations with other notions

• (a,d)-Decomposition. This is the best upper bound we know for the general case. If the edges of a graph ${\displaystyle G}$ can be partitioned into two sets, one of them inducing a graph with arboricity ${\displaystyle a}$, the second inducing a graph with maximum degree ${\displaystyle d}$, then ${\displaystyle i_{g}(G)\leq \left\lfloor {\frac {3\Delta (G)-a}{2}}\right\rfloor +8a+3d-1}$.^[28]
  If moreover ${\displaystyle \Delta (G)\geq 5a+6d}$, then ${\displaystyle i_{g}(G)\leq \left\lfloor {\frac {3\Delta (G)-a}{2}}\right\rfloor +8a+d-1}$.^[28]
• Degeneracy. If ${\displaystyle G}$ is a k-degenerated graph with maximum degree ${\displaystyle \Delta (G)}$, then ${\displaystyle i_{g}(G)\leq 2\Delta (G)+4k-2}$. Moreover, ${\displaystyle i_{g}(G)\leq 2\Delta (G)+3k-1}$ when ${\displaystyle \Delta (G)\geq 5k-1}$ and ${\displaystyle i_{g}(G)\leq \Delta (G)+8k-2}$ when ${\displaystyle \Delta (G)\leq 5k-1}$.^[27]

Graph Classes

For a class ${\displaystyle {\mathcal {C}}}$ of graphs, we denote by ${\displaystyle i_{g}({\mathcal {C}})}$ the smallest integer ${\displaystyle k}$ such that every graph ${\displaystyle G}$ of ${\displaystyle {\mathcal {C}}}$ has ${\displaystyle i_{g}(G)\leq k}$.

• Paths : For ${\displaystyle k\geq 13}$, ${\displaystyle i_{g}(P_{k})=5}$.
• Cycles : For ${\displaystyle k\geq 3}$, ${\displaystyle i_{g}(C_{k})=5}$.^[29]
• Stars : For ${\displaystyle k\geq 1}$, ${\displaystyle i_{g}(S_{2k})=3k}$.^[27]
• Wheels : For ${\displaystyle k\geq 6}$, ${\displaystyle i_{g}(W_{2k+1})=3k+2}$. For ${\displaystyle k\geq 7}$, ${\displaystyle i_{g}(W_{2k})=3k}$.^[27]
• Subgraphs of Wheels : For ${\displaystyle k\geq 13}$, if ${\displaystyle G}$ is a subgraph of ${\displaystyle W_{k}}$ having ${\displaystyle S_{k}}$ as a subgraph, then ${\displaystyle i_{g}(G)=\left\lceil {\frac {3k}{2}}\right\rceil }$.^[30]

Open Problems

• Is the upper bound ${\displaystyle i_{g}(G)<3\Delta (G)-1}$ tight for every value of ${\displaystyle \Delta (G)}$ ?^[27]
• Is the incidence game chromatic number a monotonic parameter (i.e. is it as least as big for a graph G as for any subgraph of G) ?^[27]

Notes

References (chronological order)