, On observe que toutes les chaînes dans G sont définies par des arêtes En plus, on ne peut avoir une chaîne entre s i et s j dans G (sinon, nous devrions avoir un cycle (presque) proprement-arêtes-colorées à travers s dans G c ) De cette manière, toutes les chaînes dans G commencent au sommet s i ? S k et finissent à certain sommet t j ? T k . Enfin, on construit un graphe non-arêtes-colorées G en contractons S k et T k respectivement aux sommets s et t. Notons que, les s ? t chaînes dans G sont associées aux s ? t chaînes proprement-arêtes-colorées dans G c et vice-versa. Par conséquent, nous obtenons k s ? t chaînes proprement-arêtes-colorées dans G c
, ) pour p = (n ? 1)/2. Notons qu'aucun sommet ne peut être visité plus de p fois dans G c même s'il appartient à différentes s ? t marches proprement-arêtescolorées . Pour voir cela, on considère un sommet x ? G c et une s ? t marche proprementarêtes-colorées de longueur 2 qui passe par x, p.92
, presque) proprement-arêtes-colorées qui passe à travers s dans G c . Maintenant, utilisant le théorème 5.3, nous pouvons facilement prouver que G c contient une marche (presque) proprementarêtes-colorées qui passe à travers s si, et seulement si, H c contient un (presque) proprementarêtes-colorées qui passe à travers s . Par conséquent, nous n'avons aucun cycle (presque) proprement-arêtes-colorées qui passe à travers s dans H c . Ainsi, d'après le théorème 6.8 on peut trouver (en un temps polynomial) k chaînes proprement-arêtes-colorées entre s ? Henri Michaux
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