General Antifactors of Graphs

In a superstitious company everybody has numbers that he thinks to be unlucky for himself. When they meet, everybody wants to shake hands with some of his acquaintances, hut nobody wants to shake hands with an unlucky number of acquaintances. When can this be successful? This question occurred to L. Lovلsz (Period. Moth. Hungar.4, Nos. 2-3, 1973, 121-123), where the case when everyone has one unlucky number antifactor problem) is answered. In this paper we give a "Tutte-type good characterization" (and a simple polynomial algorithm) to decide this question when several unlucky numbers are allowed. but no one in the company has two neighboring unlucky numbers.