game chromatic number

game chromatic number of G in the game Alice plays first the game Bob plays first game chromatic number of G in

Question. AliceBob 2 1 Alice ?

Question. AliceBob 1 1 Alice 2 Bob 2

Theorem. (Faigle et al.) (Kierstead and Trotter) Theorem. (Zhu) Theorem.

(Zhu)

L(p,q)-labeling

Question Since ? 0

Theorem. (Griggs and Yeh) Conjecture. (Griggs and Yeh) Theorem. (Gonçalves)

L(p,q)-labeling game

Alice plays first Bob plays first

Question. Alice Bob a 2 a +3 0 b Alice a b

Question. Alice Bob a 2 b Note.

Lemma.

Theorem

Example. Alice 0 Bob 2 Alice plays firstBob plays first 1 3

Example. Alice plays first Alice 2 Bob 0 Alice 4 Bob plays first 1 3 Bob 5

Alice Bob 2 5 Alice 0 Bob 7 Example. Alice plays first

Alice Bob a 5 Alice Bob ? b

Bob plays first 1 Alice 5

Bob plays first Bob Alice 2 b Bob Alice a c

Question.

Observation 1. Alice … 7 1 Bob 5 b 0

Observation 2. Alice … 7 Bob 1 5

Theorem vertices, 7 numbers

Question. How to prove this theorem? (Use induction, we already know that

Idea. Alice’s strategy

Idea. Bob’s strategy

Example nd … 4th 1st 6th3rd 5th 7th nd … 1st 4th 3rd 5th

Theorem.

Example

Question. Alice 017 Bob

Bob’s strategy proof. a 27 Alice Bob Alice a 15

proof. Alice’s strategy Alice 16 Bob b Alice Alice 17 Alice c Bob 11 Alice

b 22 Bob 10 proof. Alice’s strategy 16 c 19 1 Bob 2 20 Alice

Theorem.

Thanks!

Note. It is not true that if Alice(resp. Bob) plays first, and at some step, he can move twice, then the smallest number needed to complete the game is less than or equal to(resp. ). For example, but if at the first step, Alice need to move twice, then the smallest number needed to complete this game is 6({0,1,2,…,6}).