| Research Abstract |
The purpose of this research was to investigate the relationship between factors of graphs and forbidden subgraphs. During the research period, we have obtained a number of results. Some of them are listed as follows ;
1.Sumner (1974) has proved that every connected K_<1,3>-free graph of even order has a 1-factor. We investigated the converse of this type, and proved that if every connected H-free graph of even order has a 1-factor, then H is isomorphic to either K_<1,2> or K_<1,3>. This result says that essentially K_<1,3> is the only graph that forces the existence of a 1-factor by forbidding it.
2.We proved that if every connected {H_1,H_2}-graph of even order has a 1-factor, then wither H_1 or H_2 is isomorphic to K_<1,2> or K_<1,3>. This result says that if we forbid a pair of graphs to force the existence of a 1-factor, then one of the pair always becomes redundant.
3.On the other hand, we found that if we forbid three graphs, then the situation changes. We found a triple {H_1,H_2,H_3} such that every connected {H_1,H_2,H_3}-free graph of even order has a 1-factor, and none of them are isomorphic to a subgraph of K_<1,3>. This result suggests that there are a non-redundant triple of graphs which force the existence of a 1-factor. Moreover, we found infinitely many such triple.
As a summary, we completely characterized a set of forbidden subgraphs forcing the existence of a 1-factor when the order of the set is at most three.
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