. En and R. Toujours-que-le-g-nombre-et-le, nombre distinguants ludiques de GH sont inférieurs à ceux de K n K m , où n et m désignent l'ordre respectif de G et H

{. .. Démonstration and ?. {1, Pour les trois autres points, notons d'abord que puisque n = m, les facteurs K n et K m sont relativement premiers. Les sommets du graphe K n K m sont notés (i, j), avec i ?

?. Cette-méta-couleur-est-une-liste and . {1, n}, est le nombre de sommets coloriés avec la couleur l dans cette K n -bre. Il est facile, mais important, Proposition 4.45. Soient G et H deux graphes connexes relativement premiers, D(H) ? 2

|. Si and |. , H)| sont impairs

. Démonstration, Nous commençons par le premier point Appelons c la coloration construite durant la partie. Pour chaque G-bre G v , avec v ? V (H), nous dénissons : p(G v ) = 1 si |{u ? H v |c((u, v)) = 1}| est impair 2 sinon

. De-celle-ci, Gentle peut donc suivre une stratégie gagnante sur G dans ces bres Dans les autres G-bres, c'est aussi le cas, sauf pour le premier coup qui est joué par Rascal. En d'autres termes, Rascal va jouer deux coups d'alée dans ces G-bres. Puisque G est sommet-transitif, Gentle peut, d'après le Lemme 4.34, suivre également une stratégie gagnante pour G dans ces G-bres. Elle joue ainsi jusqu'à ce que l'une des G-bres soit totalement coloriée

. Puisqu-'elle-suit-une-stratégie-des-g-bres and . Rappelons-que-gentle-va-jouer-le-dernier-coup-dans-chaque-g-bre, Elle peut ainsi dans chacune d'elles décider de la parité du nombre de sommets coloriés avec 1. Démontrons maintenant que la coloration c est distinguante Soit ? un automorphisme préservant c. Pour tout v ? V (H), nous avons p(?(G v )) = p(G v ) Puisque c est une coloration distinguante de H, cela implique, d'après la propriété (4.1), que les G-bres sont stables sous l'action de ?. Nous avons donc que ?(G v 0 ) = G v 0 . Mais la coloration de cette G-bre est obtenue en suivant une stratégie gagnante pour G

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