, Now it straightforward to verify that u does not receive share according to any other rule. Furthermore, as |I(u)| = 3 and |I(u + 3)| ? 3

A. Lemma, 3 Let C be an identifying code in C n (1, 3) and u ? C be a codeword such that u does not receive share according to any of the previous rules

. Proof, Observe first that if u + 2 is a codeword, then we are immediately done since at least two of the I-sets I(u ? 1), I(u + 1) and I(u + 3) consists of at least three codewords implying s s (u) ? s(u) ? 1 + 2 · 1/2 + 2 · 1/3 = 8/3 = 11/4 ? 2/24. The same argument also applies for u ? 2 ? C. Hence, we may assume that u ? 2 and u + 2 do not belong to C. Now the proof divides into the following cases depending on the number of codewords in I(u): ? Suppose first that |I(u)| = 1, i.e., I(u) = {u}. The previous observation taken into account

, Consider first the case with I(u) = {u ? 3, u}. If now u + 4 ? C, then |I(u ? 3)| ? 3 and |I(u + 3)| ? 3 since I(u ? 3) = I(u) and I(u + 1) = I(u + 3) and we are done as s s (u) ? s(u) ? 1 + 2 · 1/2 + 2 · 1/3 = 11/4 ? 2/24. Hence, we may assume that u +4 / ? C. Therefore, as I(u ? 1) = I(u +1) = {u}, we have u ? 4 ? C. Furthermore, since I(u + 2) = ?, I(u + 1patternP

?. C-and-u-+-3-?-c, Now we have s(u) to any rule

?. C-and-u-+-4-?-c,

?. C-and-u-+-5-?-c, Observe first that u + 7 ? C since I(u + 4) = according to any rule. Moreover, at least one of u + 6, u + 8 and u + 11, say v, is a codeword since I(u + 6) = I(u + 8). , we are done as s s (u) + s s (u + 1)

?. C. , 2 · 11/4 ? 25/24 < 2 · 11/4 ? 3/4. Hence, we may assume that u + 9 / ? C and u + 10 ? C. 14/24, and u + 7 can receive share only according to the ruleR6 (3/24 units). Therefore, we are done since s s (u) + s s (u + 7) ? (11/4 ? 11/24) + (11/4 ? 14/24 + 3/24) = 2 · 11/4 ? 22/24 < 2 · 11/4 ? 3/4. Hence, we may assume that u + 11 / ? C

, For n ? 0, 1, 4 (mod 6) the result for locating-dominating codes is given in, vol.17

, Note that these patterns S3 can overlap each other. We denote by P 6 the pattern xxoooo. Therefore, we can divide overlapping S3-patterns into non-overlapping patterns P 6. Without loss of generality, we can assume that the vertices 0, vol.10, 2009.

, Alice considers an arc of P ? ( e). By Observation C.1, this happens at most 2a times. This gives at most 2a marked sons by this rule, ? By rule, vol.2

, The arc e is marked right after or already marked when this happens 2k times. This gives at most 2k marked sons by this rule, ? By rule 2(b)

, This can happen only if c > b, otherwise, by rule 2(b), e is already marked, ? By rule, vol.2

, As e is unmarked, this can happen only if d > b and at most d times. This gives at most ? d>b d marked sons by this rule, ? By rule 2(d)

M. Albert, R. J. Nowakowski, and D. Wolfe, Lessons in Play: An Introduction to Combinatorial Game Theory, 2007.

S. Aggarwal, J. Geller, S. Sadhuka, and M. Yu, Generalized Algorithm for Wythoff 's Game with Basis Vector, 2016.

S. D. Andres, The game chromatic index of forests of maximum degree ? ? 5, Discrete Applied Mathematics, vol.154, pp.1317-1323, 2006.

S. Arnborg, D. G. Corneil, and A. Proskurowski, Complexity of Finding Embeddings in a k-Tree, SIAM. Journal on Algebraic and Discrete Methods, vol.8, pp.277-284, 1987.

R. B. Austin, Impartial and Partizan Games, 1976.

T. Bartnicki, B. Bre?ar, J. Grytczuk, M. Kov?e, Z. Miechowicz et al., Game chromatic number of Cartesian product graphs, Electron. J. Combin, vol.15, issue.1, 2008.

T. Bartnicki and J. Grytczuk, A note on the game chromatic index of graphs, Graphs and Combinatorics, vol.24, pp.67-70, 2008.

Y. Ben-haim and S. Litsyn, Exact minimum density of codes identifying vertices in the square grid, SIAM J. Discrete Math, vol.19, pp.69-82, 2005.

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways For Your Mathematical Plays, 1982.
DOI : 10.1201/9780429487330

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, vol.3, 2009.

N. Bertrand, I. Charon, O. Hudry, and A. Lobstein, Identifying and Locating-dominating codes on chains and cycles, European Journal of Combinatorics, vol.25, pp.969-987, 2004.
DOI : 10.1016/j.ejc.2003.12.013
URL : https://doi.org/10.1016/j.ejc.2003.12.013

A. Beveridge, T. Bohman, A. Frieze, and O. Pikhurko, Game chromatic index of graphs with given restrictions on degrees, Theoretical Computer Science, vol.407, pp.242-249, 2008.

H. Bodlaender, On the complexity of some coloring games, Graph-Theoretic Concepts in Computer Science. WG, vol.484, 1990.

H. L. , Bodlaender A linear time algorithm for finding tree-decompositions of small treewidth, SIAM Journal on Computing, vol.25, issue.6, pp.1305-1317, 1996.

O. V. Borodin, A. O. Ivanova, A. V. Kostochka, and N. N. Sheikh, Decompositions of quadrangle-free planar graphs, Discussiones Mathematicae, Graph Theory, vol.29, pp.87-99, 2009.

L. Cai and X. Zhu, Game chromatic index of k-degenerate graphs, Journal of Graph Theory, vol.36, issue.3, pp.144-155, 2001.

C. Charpentier, B. Effantin, and G. Paris, On the game coloring index of F +-decomposable graphs, vol.236, pp.73-83, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01883462

I. Charon, O. Hudry, and A. Lobstein, Identifying codes with small radius in some infinite regular graphs, The Electronic Journal of Combinatorics, vol.9, 2002.

C. Chen, C. Lu, and Z. Miao, Identifying codes and Locating-dominating sets on paths and cycles, Discrete Applied Mathematics, vol.159, pp.1540-1547, 2011.
DOI : 10.1016/j.dam.2011.06.008
URL : https://doi.org/10.1016/j.dam.2011.06.008

G. Cohen, S. Gravier, I. Honkala, A. Lobstein, M. Mollard et al., Improved identifying codes for the grid, Electronic Journal of Combinatorics, vol.1, p.19, 1999.

G. D. Cohen, I. Honkala, and A. Lobstein, On code identifying vertices in the two dimensional square lattice with diagonals, IEEE Transactions on Computers, vol.50, issue.2, 2001.

G. D. Cohen, I. Honkala, A. Lobstein, and G. Zémor, On identifying codes, DIMACS Series in Discrete Mathematics and Theoritical Computer Science, vol.56, 2001.
DOI : 10.1090/dimacs/056/07

G. Cohen, A. Lobstein, and G. Zéemor, Identification d'une station défaillante dans un contexte radiomobile, Aspects Algorithmiques des Télécommunications (Algo-Tel '99), Actes, pp.19-22, 1999.

A. Dailly, É. Duchêne, U. Larsson, and G. Paris, Partition games are pure breaking games, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01723190

T. R. Dawson, Five Classics of Fairy Chess, 1973.

T. Dinski and X. Zhu, A bound for the game chromatic number of graphs, Discrete Mathematics, vol.196, pp.109-115, 1999.

É. Duchêne, A. S. Fraenkel, R. J. Nowakowski, and M. Rigo, Extensions and restrictions of Wythoff 's game preserving its Ppositions, Journal of Combinatorial Theory, Series A, vol.117, issue.5, pp.545-567, 2010.

É. Duchêne and S. Gravier, Geometrical extensions of Wythoff 's game, vol.309, pp.3595-3608, 2009.

P. L. Erdös, U. Faigle, W. Hochstâttler, and W. Kern, Note on the game chromatic index of trees, Theoretical Computer Science, vol.313, pp.371-376, 2004.

P. Erdös and J. L. Selfridge, On a combinatorial game, Journal of Combinatorial Theory, Series A, vol.14, issue.3, pp.298-301, 1973.

G. Exoo, Computational results on identifying t-codes, 1999.

U. Faigle, W. Kern, H. Kierstead, and W. T. Trotter, On the game chromatic number of some classes of graphs, Ars combinatoria, vol.35, pp.143-150, 1993.

A. Flammenkamp,

A. S. Fraenkel and I. Borosh, A generalization of Wythoff 's game, Journal of Combinatorial Theory, Series A, vol.15, issue.2, pp.175-191, 1973.

A. S. Fraenkel, How to Beat Your Wythoff Games' Opponent on Three Fronts, The American Mathematical Monthly, vol.89, issue.6, pp.353-361, 1982.

M. Gardner, Mathematical Games, vol.23, 1981.

M. Ghebleh and L. Niepel, Locating and identifying codes in circulant networks, Discrete Applied Mathematics, vol.161, 2001.

S. W. , Golomb A mathematical investigation of games of "take-away, Journal of Combinatorial Theory, vol.1, pp.443-458, 1966.

D. Gonçalves, Covering planar graphs with forests, one having bounded maximum degree, Journal of Combinatorial Theory, Series B, vol.99, pp.314-322, 2009.

S. Gravier, J. Moncel, and A. Semri, Identifying codes of cycles, European Journal of Combinatorics, vol.27, pp.767-776, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00360168

D. J. Guan and X. Zhu, Game chromatic number of outerplanar graphs, Journal of Graph Theory, vol.30, issue.1, pp.67-70, 1999.

J. P. Grossman and R. J. Nowakowski, A ruler regularity in hexadecimal games, Games of, vol.63, 2015.

R. K. Guy and C. A. Smith, /The G-values of various games, Mathematical Proceedings of the Cambridge Philosophical Society, vol.52, pp.514-526, 1956.

A. Hales and R. Jewett, Regularity and positional games, Trans. Amer. Math. Soc, vol.106, pp.222-229, 1963.

W. He, X. Hou, K. Lih, J. Shao, W. Wang et al., Edge-partitions of Planar Graphs and Their Game Coloring Numbers, Journal of Graph Theory, vol.41, issue.4, pp.307-317, 2002.

I. Honkala, An optimal Locating-dominating set in the infinite triangular grid, Discrete Mathematics, vol.306, pp.2670-2681, 2006.

I. Honkala and T. Laihonen, On locating-dominating sets in infinite grids, European Journal of Combinatorics, vol.27, pp.218-227, 2006.

I. Honkala and T. Laihonen, On a new class of identifying codes in graphs, Information Processing Letters, vol.102, pp.92-98, 2007.

S. Howse and R. J. Nowakowski, Periodicity and arithmetic-periodicity in hexadecimal games, Theoretical Computer Science, vol.313, issue.3, pp.463-472, 2004.

V. Junnila and T. Laihonen, Optimal identifying codes in cycles and paths, Graphs and Combinatorics, vol.28, pp.469-481, 2012.
DOI : 10.1007/s00373-011-1058-6

V. Junnila and T. Laihonen, Collection of codes for tolerant location, Proceedings of the Bordeaux Graph Workshop, pp.176-179, 2016.

V. Junnila, T. Laihonen, and G. Paris, Optimal bounds on codes for location in circulant graphs, 2018.

V. Junnila, T. Laihonen, and G. Paris, Solving Two Conjectures regarding Codes for Location in Circulant Graphs, 2017.

M. G. Karpovsky, K. Chakrabarty, and L. B. Levitin, On a new class of codes for identifying vertices in graphs, IEEE Transactions on Information Theory, vol.44, issue.2, pp.599-611, 1998.

H. A. Kierstead, A Simple Competitive Graph Coloring Algorithm, Journal of Combinatorial Theory, Series B, vol.78, issue.1, pp.57-68, 2000.

H. A. Kierstead and W. T. , Trotter Planar graph coloring with an uncooperative partner, Journal of Graph Theory, vol.18, pp.569-584, 1994.

H. A. Kierstead and D. Yang, Asymmetric marking game on line graphs, Discrete Mathematics, vol.308, issue.9, pp.1751-1755, 2008.

P. C. Lam, W. C. Shiu, and B. Xu, Edge game coloring of graphs Graph Theory, vol.37, pp.17-19, 1999.

S. Liu and H. Li, A class of extensions of Restricted (s, t)-Wythoff 's game, Open Mathematics, vol.15, issue.1, 2017.

M. Montassier, P. Ossona-de-mendez, A. Raspaud, and X. Zhu, Decomposing a graph into forests, Journal of Combinatorial Theory, Series B, vol.102, pp.38-52, 2012.
URL : https://hal.archives-ouvertes.fr/lirmm-01264344

C. Nash-williams, Decomposition of Finite Graphs Into Forests, Journal of the London Mathematical Society, pp.12-12, 1964.

D. F. Rall and P. J. Slater, On location-domination numbers for certain classes of graphs, Cong. Num, vol.45, 1984.

D. L. Roberts and F. S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics, vol.29, issue.1, pp.72-82, 2008.

Y. Sekiguchi, The game coloring number of planar graphs with a given girth, Discrete Mathematics, vol.330, pp.11-16, 2014.

C. Sia, Game chromatic number of some families of cartesian products of graphs, AKCE International Journal of Graphs and Combinatorics, vol.6, 2009.

E. Sidorowicz, Game chromatic number, game coloring number of cactuses, Information Processing Letters, vol.102, issue.4, pp.147-151, 2007.

A. N. Siegel, Combinatorial game theory, 2013.

P. J. Slater, Fault-tolerant locating-dominating sets, Discrete Mathematics, vol.249, pp.179-189, 2002.
DOI : 10.1016/s0012-365x(01)00244-8
URL : https://doi.org/10.1016/s0012-365x(01)00244-8

K. Wagner, Über eine Eigenschaft der ebenen Komplexe, Mathematische Annalen, vol.114, issue.1, pp.570-590, 1937.
DOI : 10.1007/bf01594196

Y. Wang and Q. Zhang, Decomposing a planar graph with gith at least 8 into a forest and a matching, Discrete Mathematics, vol.311, pp.844-849, 2011.
DOI : 10.1016/j.disc.2011.01.019
URL : https://doi.org/10.1016/j.disc.2011.01.019

J. Wu and X. Zhu, Lower bounds for the game colouring number of partial k-trees and planar graphs, Discrete Mathematics, vol.308, pp.2637-2642, 2008.

J. Wu and X. Zhu, Lower bounds for the game coloring number of partial k-trees and planar graphs, Discrete Mathematics, vol.308, pp.2637-2642, 2008.

W. A. Wythoff, A modification of the game of Nim, Nieuw Arch. Wisk, vol.7, issue.2, pp.199-202, 1907.

M. Xu, K. Thulasiraman, and X. Hu, Identifying codes of cycles with odd orders, European Journal of Combinatorics, vol.29, pp.1717-1720, 2008.

X. Zhu, Game coloring number of Planar graphs, Journal of Combinatorial Theory, Series B, vol.75, issue.2, pp.245-258, 1999.

X. Zhu, The game coloring number of pseudo partial k-trees, Discrete Mathematics, vol.215, pp.245-262, 2000.
DOI : 10.1016/s0012-365x(99)00237-x
URL : https://doi.org/10.1016/s0012-365x(99)00237-x

X. Zhu, Game coloring the Cartesian product of graphs, J. Graph Theory, vol.59, pp.261-278, 2008.