Graph coloring game

Acyclic coloring. Every graph ${\displaystyle G}$ with acyclic chromatic number ${\displaystyle k}$ has ${\displaystyle \chi _{g}(G)\leq k(k+1)}$.[4]

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 ${\displaystyle \chi _{g}(G)\leq 4+c}$.[5]

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

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 ${\displaystyle n,m}$.[16]

• ${\displaystyle \chi _{g}(K_{2}\square P_{1})=2}$,
  ${\displaystyle \chi _{g}(K_{2}\square P_{k})=3}$ if ${\displaystyle k\in \{2,3\}}$,
  and ${\displaystyle \chi _{g}(K_{2}\square P_{k})=4}$ if ${\displaystyle k\geq 4}$.[16]
• For any ${\displaystyle k\geq 3}$, ${\displaystyle \chi _{g}(K_{2}\square C_{k})=4}$.[16]
• For any ${\displaystyle k}$, ${\displaystyle \chi _{g}(K_{2}\square K_{k})=k+1}$.[16]
• Trees : ${\displaystyle \chi _{g}(T_{1}\square T_{2})\leq 12}$.
• Stars : ${\displaystyle \chi _{g}(S_{m}\square P_{1})=2}$, ${\displaystyle \chi _{g}(S_{m}\square P_{2})=3}$,
  and ${\displaystyle \chi _{g}(S_{m}\square P_{n})=\chi _{g}(S_{m}\square C_{n})=4}$ if ${\displaystyle n\geq 3}$.[17]
• Wheels : ${\displaystyle \chi _{g}(P_{2}\square W_{n})=5}$ if ${\displaystyle n\geq 9}$.[17]
• Complete bipartite graphs : ${\displaystyle \chi _{g}(P_{2}\square K_{m,n})=5}$ if ${\displaystyle m,n\geq 5}$.[17]

These questions are still open to this date.

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.[19] There exists graphs with ${\displaystyle \chi '_{g}(G)>1.008\Delta (G)}$ for arbitrary large values of ${\displaystyle \Delta (G)}$.[20]

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 ${\displaystyle G}$.[20]

• Arboricity. Let ${\displaystyle a(G)}$ be the arboricity of a graph ${\displaystyle G}$. Every graph ${\displaystyle G}$ with maximum degree ${\displaystyle \Delta (G)}$ has ${\displaystyle \chi '_{g}(G)\leq \Delta (G)+3a(G)-1}$.[21]

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 ${\displaystyle n\geq 4}$.[19]
• Forests : ${\displaystyle \chi '_{g}({\mathcal {F}}_{\Delta })\leq \Delta +1}$ when ${\displaystyle \Delta \neq 4}$, and ${\displaystyle 5\leq \chi '_{g}({\mathcal {F}}_{4})\leq 6}$.[22]
  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 ${\displaystyle \chi '_{g}(F)\leq 5}$.[23]

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 ?[19]

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 ${\displaystyle \chi '_{g}(G)\geq (1+\epsilon )\delta (G)}$. [20]

• (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 ${\displaystyle i_{g}(G)\leq \left\lfloor {\frac {3\Delta (G)-a}{2}}\right\rfloor +8a+3d-1}$.[25]
  If moreover ${\displaystyle \Delta (G)\geq 5a+6d}$, then ${\displaystyle i_{g}(G)\leq \left\lfloor {\frac {3\Delta (G)-a}{2}}\right\rfloor +8a+d-1}$.[25]
• 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 ${\displaystyle \Delta (G)\leq 5k-1}$.[24]

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 ${\displaystyle k\geq 3}$, ${\displaystyle i_{g}(C_{k})=5}$.[26]
• Stars : For ${\displaystyle k\geq 1}$, ${\displaystyle i_{g}(S_{2k})=3k}$.[24]
• Wheels : For ${\displaystyle k\geq 6}$, ${\displaystyle i_{g}(W_{2k+1})=3k+2}$. For ${\displaystyle k\geq 7}$, ${\displaystyle i_{g}(W_{2k})=3k}$.[24]
• 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 ${\displaystyle i_{g}(G)=\left\lceil {\frac {3k}{2}}\right\rceil }$.[27]

• Is the upper bound ${\displaystyle i_{g}(G)<3\Delta (G)-1}$ tight for every value of ${\displaystyle \Delta (G)}$ ?[24]
• 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) ?[24]