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Консультация № 137574
19.05.2008, 22:39
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Здравствуйте! Вопрос по теории графов. Решал задачу, в которой нужно найти ейлеров и гамильтонов циклы. Я знаю, что цикл. ейлеров путь, это тот который проходит по всем ребрам в точности по одному разу и возвр. в начальную вершину. Я доказал, что циклического ейлерового не существует т.к. степень почти всех вершин - непарная. и что не существует не цикл. ейл. путь (по одной теореме). Подскажите как доказать, что не существует или существует цикл. или нецикл. гамильтонов маршрут. Просто теорию вкратце. Спасибо заранее.
Обсуждение
21.05.2008, 00:33
общий
это ответ
Здравствуйте, Sashka!
Между эйлеровыми и гамильтоновыми линиями имеется определенная аналогия. Эйлерова линия проходит один раз по каждому ребру, а гамильтонова – через каждую вершину. Но существует и большая разница.
Эилер доказал, что если степени всех вершин связного графа четны, то на таком графе можно найти цикл, содержащий все ребра этого графа, причем в точности по одному разу. Для гамильтонова цикла общего критерия, насколько мне известно, не существует. Поэтому для того, чтобы доказать, что на конкретном графе гамильтонов цикл существует, надо его найти…
Существуют лишь некоторые критерии для частных случаев. Например, если для любой пары вершин графа сумма локальных степеней больше или равна числу вершин графа, то в таком графе существует гамильтонов цикл… Полагаю, что специалисты по дискретной математике дадут более исчерпывающие справки.
С уважением.
Между эйлеровыми и гамильтоновыми линиями имеется определенная аналогия. Эйлерова линия проходит один раз по каждому ребру, а гамильтонова – через каждую вершину. Но существует и большая разница.
Эилер доказал, что если степени всех вершин связного графа четны, то на таком графе можно найти цикл, содержащий все ребра этого графа, причем в точности по одному разу. Для гамильтонова цикла общего критерия, насколько мне известно, не существует. Поэтому для того, чтобы доказать, что на конкретном графе гамильтонов цикл существует, надо его найти…
Существуют лишь некоторые критерии для частных случаев. Например, если для любой пары вершин графа сумма локальных степеней больше или равна числу вершин графа, то в таком графе существует гамильтонов цикл… Полагаю, что специалисты по дискретной математике дадут более исчерпывающие справки.
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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