Game chromatic number of graphs Two players Alice
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 Or there are uncolored vertices, but there is no legal color for any of the uncolored vertices Bob wins Alice 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.
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.
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,
Theorem [Guan and Zhu, J. Graph Theory, 1999] For any outerplanar graph G, c g ( G ) £ 7.
Theorem [Zhu, Discrete Math. , 2000] For any partial k-tree G, c g ( G ) £ 3 k+2. 2 -tree A partial k-tree is a subgraph of a k-tree
Theorem [Zhu] For any planar graph G, c g ( G ) £ 17.
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, Proof: There is a forest F such that
Theorem [Guan and Zhu, J. Graph Theory, 1999] For any outerplanar graph G, c g ( G ) £ 7. Proof: G’
Theorem [Guan and Zhu, J. Graph Theory, 1999] For any outerplanar graph G, c g ( G ) £ 7. Proof: 7 2 10 12 1 11 3 4 5 6 8 9 G’
1 2 3
1 2 3 4 6 7 5 8
1 3 4 9 17 22 5 6 7 11 2 13 8 10 12 14 15 18 19 23 24 16 20 21
For each uncolored vertex v, there at most 3 colored neighbours in T.
1 3 4 9 17 22 5 6 7 11 For each uncolored vertex v, there at most 6 colored neighbours in G’. 2 13 8 10 12 14 15 18 19 23 24 16 20 21
1 3 4 9 17 22 5 6 7 11 For each uncolored vertex v, there at most 6 colored neighbours in G. 2 13 8 10 12 14 15 18 19 23 24 16 20 21
Theorem [Guan and Zhu, J. Graph Theory, 1999] For any outerplanar graph G, c g ( G ) £ 7. There is an outerplanar G such that
Theorem [Zhu, Discrete Math. , 2000] For any partial k-tree G, c g ( G ) £ 3 k+2. Proof: 1 2 k+1 k+2 k k+3 n
Theorem [Zhu, Discrete Math. , 2000] For any partial k-tree G, c g ( G ) £ 3 k+2. Proof: k 1 2 k k n
Theorem [Zhu, Discrete Math. , 2000] For any partial k-tree G, c g ( G ) £ 3 k+2. Proof: 1 2
Theorem [Zhu, Discrete Math. , 2000] For any partial k-tree G, c g ( G ) £ 3 k+2. Proof: 1 2
Theorem [Zhu, Discrete Math. , 2000] For any partial k-tree G, c g ( G ) £ 3 k+2. Proof: Uncolored vertex x
Theorem [Zhu, Discrete Math. , 2000] For any partial k-tree G, c g ( G ) £ 3 k+2. Proof: Uncolored vertex x
Theorem [Zhu, Discrete Math. , 2000] For any partial k-tree G, c g ( G ) £ 3 k+2. Proof: Uncolored vertex x
Theorem [Zhu, Discrete Math. , 2000] For any partial k-tree G, c g ( G ) £ 3 k+2. Proof: Uncolored vertex x There at most k +2(k+1) colored neighbours of x. k +2 k+1=3 k+1
t colors game coloring number game chromatic number game coloring number
Game chromatic number Author Graph Upper bound Forests 4 Guan and Zhu Outerplanar 7 Faigle, Kern, Kierstead, Trotter Interval graphs 3 k+1 Zhu Partial k-tree 3 k+2 Zhu Planar 17 Faigle, Kern, Kierstead, Trotter Game coloring number Graph Lower bound Upper bound Forests 4 4 outerplanar 7 7 Interval graphs 3 k+1 Partial k-tree 3 k+2 Planar 11 17
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