2002 Fiscal Year Final Research Report Summary
Research on Factors of Graphs Using Closure Operations
| Project/Area Number |
13640138
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | NIHON UNIVERSITY |
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Principal Investigator |
SAITO Akira Nihon Univ., Dept. Computer Science and System Analysis, Professor, 文理学部, 教授 (90186924)
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| Project Period (FY) |
2001 – 2002
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| Keywords | closure / factor / degree-sum / Ryjacek closure / factor-critical / extendable |
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| Research Abstract |
The purpose of this research was to apply closure operations, which had been used in the study of hamiltonian cycles, to factor theory and get new insights. We obtained the following results.
・ We obtained a new degree-sum condition for a bipartete graph to have a 2-factor containing a specified edge.
・ We obtained a deep insight on the structure of Ryjacek closure for claw-free graphs. In particular, we proved that if the Ryjacek closure of a claw-free graph G of order n is a complete graph, then G has a cycle of length n - 1.
・ We obtained a degree-sum condition for a graph G to satisfy diff(G) 【less than or equal】 1, where diff(G) is the difference between the order of a longest path and the order of a longest cycle in G.
・ We obtained a new degree-sum condition for a k-factor-critical graph to be hamiltonian.
・ By extending the above result, we obtained a new degree-sum condition for a k-extendable graph to be hamiltonian.
・ We gave a new necessary and sufficient condition for a graph to be 1-extendable in terms of alternating paths. Using this result, we gave a new algorithm to check 1-extendability of a graph G, which runs in O(|E(G)|^2).
・ For given positive integers k and n, we determined the class of graphs such that every cycle has length k (mod n). We proved that this class has a rich structure only if k=0 and n=2, i.e. the class of bipartite graphs.
・ We proved that the Ryjacek closure can be a useful tool in the study of k-factor-critical graphs.
・ We obtained a new degree-sum condition for a bipartite graph to have a 2-factor containing specified vertices in each component. Although the problem is similar to the one at the top of this list, the conclusion was entirely different.
As we see above, we obtained a number of interesting results, and hence the research was successful.
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