Research on Factors of Graphs Using Closure Operations
Project/Area Number 
13640138

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  NIHON UNIVERSITY 
Principal Investigator 
SAITO Akira Nihon Univ., Dept. Computer Science and System Analysis, Professor, 文理学部, 教授 (90186924)

Project Period (FY) 
2001 – 2002

Project Status 
Completed (Fiscal Year 2002)

Budget Amount *help 
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2002: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2001: ¥1,600,000 (Direct Cost: ¥1,600,000)

Keywords  closure / factor / degreesum / Ryjacek closure / factorcritical / extendable / ハミルトン閉路 / 拡張可能性 / グラフ / 2因子 / 閉路 / 最長閉路 / 最長パス / 内周 
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 degreesum condition for a bipartete graph to have a 2factor containing a specified edge. ・ We obtained a deep insight on the structure of Ryjacek closure for clawfree graphs. In particular, we proved that if the Ryjacek closure of a clawfree graph G of order n is a complete graph, then G has a cycle of length n  1. ・ We obtained a degreesum 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 degreesum condition for a kfactorcritical graph to be hamiltonian. ・ By extending the above result, we obtained a new degreesum condition for a kextendable graph to be hamiltonian. ・ We gave a new necessary and sufficient condition for a graph to be 1extendable in terms of alternating paths. Using this result, we gave a new algorithm to check 1extendability 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 kfactorcritical graphs. ・ We obtained a new degreesum condition for a bipartite graph to have a 2factor 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.

Report
(3 results)
Research Products
(18 results)