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The game L(d ,1) - labeling problem of graphs game chromatic number Given a graph G and a set X of colors. Consider t he two - person game played on G: Alice and Bob alternate turns. A move consisting of selecting an uncolored vertex v of G and assigning it a color from X distince from the color assigned previously to neighbors of v. Alice wins if all the vertices are successful ly labeled, else Bob wins. game chromatic number of G least cardinalit y of a color set X for which Alice has a winning strategy (G) : game chromatic number of G in 1 g the game Alice plays first (G) : game chromatic number of G in 2 g the game Bob plays first Question. (C4 ) ? 1 g Alice 1 Alice ? 2 Bob (C4 ) 2 1 g Since (G) (G) 1, we also have i g (C4 ) 3 1 g (C4 ) 3 1 g Question. (C4 ) ? 2 g Alice 1 1 Bob Alice 2 2 Bob (C4 ) 2 2 g Since (G) (G), we also have i g (C4 ) 2 2 g (C4 ) 2 2 g Theorem. (Faigle et al.) The game chromatic number of the class of forests is 4. Theorem. (Kierstead and Trotter) The game chromatic number of the class of planar graph is between 8 and 33. Theorem. (Zhu) The game chromatic number of the class of planar graph is between 8 and 17. Theorem. (Zhu) The game chromatic number of the class of outerplana r graph is between 6 and 7. L(p,q)-labeling L( p, q) - labeling of G a function f from V (G ) to nonnegativ e integers satisfying (1) | f (u) f (v ) | p if d (u, v ) 1 (2) | f (u) f (v ) | q if d (u, v ) 2 k - L( p, q) - labeling of G an L( p, q) - labeling such that no label is greater th an k L( p, q) - labeling number of G p ,q (G ) : smallest number k , such that G has a k - L( p, q) - labeling (when q 1, we use p (G ) to replace p ,q (G )) (C4 ) ? Question. Since 0 23 4 ?1 ((C 4 f is a 3 - L 2,14 ) -labeling of C4 (C4 ) 4 the four verti ces are labeled by 0,1,2,3 one vertex is labeled by 2 (C4 ) 4 Theorem. (Griggs and Yeh) The L(2,1) - labeling problem is NP - complete for general graphs. Conjecture. (Griggs and Yeh) For any graph G with (G) 2, (G) (G). 2 Theorem. (Gonçalves) For any graph G, (G ) (G ) (G ) 2. 2 game L(p,q)-labeling Given a graph G and nonnegativ e integers k , p, q, p q 0. Consider t he two - person game played on G : Alice and Bob alternate turns. A move consisting of selecting an unlabeled vertex v of G and label it with a number a in {0,1,2,..., k} satisfying the conditions that (1) If u is labeled by b and d (u, v ) 1, then |b-a| p. (2) If u is labeled by b and d (u, v ) 2, then |b-a| q. Alice wins if all the vertices are successful ly labeled, else Bob wins. game L( p, q) - labeling number of G smallest k , so that Alice has a winning strategy ~ p ,q 1 (G ) : Alice plays first ~ p ,q 2 (G ) : Bob plays first ~2 Question. 1 (C4 ) ? Alice a 2 a 0 Bob b a a+3 2 b a 1 or 6 1 (C 4 ) 6 5 4 Bob Alice ~ 2 Alice can ~complete this game using 2 1 (in C 4{)0 the numbers ,1,62,...,6} ~ 2 Question. Alice 2 (C4 ) ? a b 1 a 2 b Bob ~ 2 5 4this game using 2 (C 4 ) Alice cancomplete Note. the numbers in {0,1,2,...,5} ~ 2 ~ 2 1 (G) 2 (G) dose not hold in general! Lemma. For any graph G , ~ d i (G ) d (G ) (i 1,2) Lemma. If G is a graph with maximum degree , then ~ d i (G ) ( 2d 2) for i 1,2 and d 2. Theorem. ( K n ) 2n 2 0 2 4 6 Example. Alice Bob plays first Alice 30 ~ 2 12 12 ( K 22) 23 Bob Example. Alice Bob plays playsfirst first Alice Alice Bob 32 54 10 ~ 2 12 ( K 3 ) 45 Bob Example. Alice plays first Alice 2 0 Alice Bob 5 7 Bob ~ 2 1 ( K 4 ) 7 b 3 Alice a 3 a b Alice Bob 5 ? Bob ~ 2 1 ( K 4 ) 6 ~ 2 1 ( K 4 ) 7 Bob plays first Alice 5 Bob 1 ~ 2 1 ( K 3 ) 4 ~ 2 ~ 2 2 ( K 4 ) 1 ( K 3 ) 3 7 Alice Bob plays first a 0 or b 2 a Alice 2 b 4 Bob b c ~ 2 2 ( K 4 ) 7 ~ 2 2 ( K 4 ) 7 Bob Question. ~ 2 1 ( K n ) ? ~ 2 2 ( K n ) ? Observation 1. ~ 2 ~ 2 1 ( K n ) 2 ( K n 2 K 1 ) 4 Alice 2 Bob b 0 Bob 1 5 0 1 2 3 4 5 6 7 … Observation 2. ~ 2 ~ 2 2 ( K n ) 1 ( K n1 ) 3 2 5 Bob Alice 1 0 1 2 3 4 5 6 7 … Theorem. 7n 9 1 ( K n ) 3 ~ 2 7n 7 2 ( K n ) 3 ~ 2 0 1 2 3 4 3 vertices, 7 numbers 5 6 7n 3 Question. How to prove this theorem? (Use induction, we already know that ~ 2 ~ 2 ~ 2 ~ 2 1 ( K n ) 2 ( K n 2 K 1 ) 4, 2 ( K n ) 1 ( K n1 ) 3) Idea. Alice’s strategy If v is still unlabeled, choose a smallest i , so that all the numbers in {i , i 1, i 2} can be used to label v. Labels v with i 2. If no such i exists, choose a number j that can be used to label v. Labels v with j. Idea. Bob’s strategy If v is still unlabeled, choose a smallest i , so that all the numbers in {i , i 1} can be used to label v. Labels v with i 1. If no such i exists, choose a number j that can be used to label v. Labels v with j. ~ 2 1 ( K14 ) Example. 2nd 4th 6th 3rd 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 … 7th 5th ~ 2 1st 1 ( K11 K1 ) 2nd 4th 0 1 2 3 4 5 6 7 8 9 3rd 10 11 12 13 14 15 … 5th 1st Theorem. For all n 1, r 0, ~ d 1 ( K n rK 1 ) and n 3 (4d 1) [( n 1 ) mod 3 ] d 3 ~ d 2 ( K n rK 1 ) n 1 (4d 1) 2 d [( n 2 ) mod 3 ] d 3 Theorem. For all d , m, n, m n 1, d ( K m ,n ) d m n 2 11 ( K 7,5 ) ? Example. 0 17 18 19 20 21 1 2 3 4 5 6 11 ( K 7,5 ) 7 5 (11 1) 1 21 ~ 11 Question. 1 ( K 7 , 5 ) ? Y 0 17 Alice Bob X can' t use the numbers in {7,...,16} {18,...,27} ~ 11 guess : 1 ( K 7 ,5 ) 11 ( K 7 , 5 ) 11 ~ 11 1 ( K 7 , 5 ) 31 Bob’s strategy a 16 15 a proof. Alice 27 4 Bob Alice Bob a 15 16 10 21 a 17 14 ~ 11 less than 5 numbers can be used to less than 71numbers ( K 7 , 5 ) can 31be used to label these vertices label these vertices Y X ~ 11 proof. Alice 1 ( K 7 ,5 ) 32 Alice’s strategy c 4 30 28 b 11 7, if 11 b 14, c Alice b 11 7, if 18 b 21 16 29 Alice Bob Alice Bob 17 11 12 15 b Y X (11 14, or bb10 22 game18using b the 21) Alice can complete this allallnumbers the the thenumbers numbers numbers {can 0,29 132 ,be 2,..., ,}3used ,4 32 }Bob }can can to labels bebe a in {0in,0in 1,{,1228 ,..., if used usedlabel toXlabel label these these these vertices vertices vertices vertex into at the second step ~ 11 1 ( K 7 ,5 ) 32 Alice’s strategy proof. Alice 16 2 1 b Bob Y (0 b( , or 29528 27 (43 bb ))b 32) Bob Alice 20 c 19 22 10 X b 11 7, if 0 b 3, Alice can completec this game using the b,,29 ,11 74 , }} ifcan 29be be b 32. 11 inin ~ allallthe the numbers numbers { 28 { 0 1 2 ,..., , 3 , 32 can the numbers 0,1 can be used to numbers in1{0,(1K ,27,..., 32 } if Bob labels a ) 32 ,5 used used to to label label these these vertices vertices label these vertices vertex in Y at the second step Theorem. For all d , m, n, d m n 2, ~ d 1 ( K m ,n ) 2d m n 2 m ,n (where m,n [( m n) mod 2]) 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 to2 complete ~ the game is less than or equal to 1 (G ) (resp. ~ 2 ~ 2 2 (G )). For example, 1 (2C4 K1 ) 5, 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}).