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URN etd-0124111-164137 Author Ting-pang Chang Author's Email Address changdb@mail.math.nsysu.edu.tw Statistics This thesis had been viewed 5075 times. Download 957 times. Department Applied Mathematics Year 2010 Semester 1 Degree Ph.D. Type of Document Language English Title The hamiltonian numbers of graphs and digraphs Date of Defense 2011-01-19 Page Count 59 Keyword hamiltonian number hamiltonian cycle double loop network Abstract The hamiltonian number problem is a generalization of hamiltonian cycle problem in graph theory. It is well known that the hamiltonian cycle problem in graph theory is NP-complete [16]. So the hamiltonian number problem is also NP-complete. On the other hand, the hamiltonian number problem is the traveling salesman problem with each edge having weight 1.

A hamiltonian walk of a graph G is a closed spanning walk of minimum length. The length of a hamiltonian walk in G is called the hamiltonian number. For graphs, we give some bounds for hamiltonian numbers of graphs. First, we improve some results in [14] and give a necessary and sufficient condition for h(G) < e(G) where e(G) is the minimum length of a closed walk passing through all edges of G. Next, we prove that if two nonadjacent vertices u and v satisfying that deg(u)+deg(v) ≥ |G|, then h(G) = h(G + uv). This result generalizes a theorem of Bondy and Chv′atal for the hamiltonian cycle. Finally, we show that if 0 ≤ k ≤ n − 2 and G is a 2-connected graph of order n satisfying deg(u) + deg(v) + deg(w) ≥ 3n−k−2 for every independent set {u, v,w} of three vertices in G, then h(G) ≤ n+k. It is a generalization of a Bondy’s result.

For digraphs, we give some bounds for hamiltonian numbers of digraphs first. We prove that if a digraph D of order n is strongly connected, thenn ≤ h(D) ≤ ⌊(n+1)^2/4 ⌋. Next, we also present some digraphs of order n ≥ 5 which have hamiltonian number k for n ≤ k ≤ ⌊(n+1)^2/4 ⌋. Finally, we study hamiltonian numbers of M‥obius double loop networks. We introduce M‥obius double loop network and every strongly connected double loop network is isomorphic to some M‥obius double loop network. Next, we give an upper bound for the hamiltonian numbers of M‥obius double loop networks. Then, we find some necessary and sufficient conditions for M‥obius double loop networks MDL(d, m, ℓ) to have hamiltonian numbers dm, dm + 1 or dm + 2.Advisory Committee Ko-Wei Lih - chair

Dah-Jyh Guan - co-chair

Gerard Jennhwa Chang - co-chair

Xuding Zhu - co-chair

Sen-Peng Eu - co-chair

Tsai-Lien Wong - co-chair

Hong-Gwa Yeh - co-chair

Li-Da Tong - advisor

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etd-0124111-164137.pdf Date of Submission 2011-01-24