Game chromatic number of graphs 吳佼佼 中研院數學所

Two players: Alice and Bob A graph G A set of colors

Adjacent vertices cannot be colored by the same color.

Game over !

Game is over when one cannot make a move. Either all the vertices are colored Alice wins Or there are uncolored vertices, but there is no legal color for any of the uncolored vertices Bob wins

Alice’s goal: have all the vertices colored. Bob’s goal: to have an uncolored vertex with no legal color.

In the previous example, Bob wins the game ! But, Alice could have won the game if she had played carefully ! If both players play “perfectly”, who will win the game ?

It depends on the graph G, and depends on the number of colors ! Given a graph G, the game chromatic number of G is the least number of colors for which Alice has a winning strategy.

A hint that the problem is difficult. To prove that one needs to prove the correctness of a sentence of the form: MA: a move for Alice; MB: a move for Bob A hint that the problem is difficult.

Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ] H.L. Bodlaender, On the complexity of some coloring games, Computer Science, 1991. Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ] For any forest F,

7. ) ( £ c Theorem [Guan and Zhu, J. Graph Theory, 1999] For any outerplanar graph G, G 7. ) ( £ g c

3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000] G 3k+2. ) ( £ g c For any partial k-tree G, 2-tree A partial k-tree is a subgraph of a k-tree

Theorem [Zhu] For any planar graph G, G 17. ) ( £ g c

Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ] For any forest F, Proof:

Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ] For any forest F, Proof:

Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ] For any forest F, Proof:

Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ] For any forest F, There is a forest F such that Proof:

7. ) ( £ c G’ Theorem [Guan and Zhu, J. Graph Theory, 1999] For any outerplanar graph G, G 7. ) ( £ g c Proof: G’

Theorem [Guan and Zhu, J. Graph Theory, 1999] For any outerplanar graph G, G 7. ) ( £ g c Proof: 7 2 10 12 G’ 1 11 3 9 4 8 5 6

1 2 3

1 2 4 3 5 6 7 8

1 2 3 4 5 6 7 8 9 10 11 12 14 13 15 16 17 18 19 20 21 22 23 24

For each uncolored vertex v, there are at most 3 colored neighbours in T.

1 2 3 4 5 For each uncolored vertex v, there are at most 6 colored neighbours in G’. 1 2 3 4 5 6 7 8 9 10 11 12 14 13 15 16 17 18 19 20 21 22 23 24

1 2 3 4 5 For each uncolored vertex v, there are at most 6 colored neighbours in G. 1 2 3 4 5 6 7 8 9 10 11 12 14 13 15 16 17 18 19 20 21 22 23 24

7. ) ( £ c Theorem [Guan and Zhu, J. Graph Theory, 1999] For any outerplanar graph G, G 7. ) ( £ g c There is an outerplanar G such that

3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000] For any partial k-tree G, G 3k+2. ) ( £ g c Proof: 1 2 n k+1 k+2 k+3 k

3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000] For any partial k-tree G, G 3k+2. ) ( £ g c Proof: k k k k k 1 2 n

3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000] For any partial k-tree G, G 3k+2. ) ( £ g c Proof: 1 2

3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000] For any partial k-tree G, G 3k+2. ) ( £ g c Proof: 1 2

3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000] For any partial k-tree G, G 3k+2. ) ( £ g c Proof: Uncolored vertex x

3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000] For any partial k-tree G, G 3k+2. ) ( £ g c Proof: Uncolored vertex x

3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000] For any partial k-tree G, G 3k+2. ) ( £ g c Proof: Uncolored vertex x

There are at most k +2(k+1) colored neighbours of x. Theorem [Zhu, Discrete Math.,2000] For any partial k-tree G, G 3k+2. ) ( £ g c Proof: Uncolored vertex x There are at most k +2(k+1) colored neighbours of x. k +2k+1=3k+1

t colors game coloring number game chromatic number game coloring number

Game chromatic number Author Graph Upper bound Faigle, Kern, Kierstead, Trotter Forests 4 Guan and Zhu Outerplanar 7 Interval graphs 3k+1 Zhu Partial k-tree 3k+2 Planar 17 Game coloring number Graph Lower bound Upper bound Forests 4 outerplanar 7 Interval graphs 3k+1 Partial k-tree 3k+2 Planar 11 17

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