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Mutually Independent Hamiltonian Connectivity of (n,k)-Star Graphs
Selina Yo-Ping Chang1, Justie Su-Tzu Juan1, Cheng-Kuan Lin2, Jimmy J. M. Tan2, and Lih-Hsing Hsu3
1Department of Computer Science and Information Engineering, National Chi Nan University, Nantou 54561, Taiwan
2Department of Computer Science, National Chiao Tung University, Hsinchu 30010, Taiwan
3Department of Computer Science and Information Engineering, Providence University, Taichung 43301, Taiwan
lhhsu@pu.edu.tw
Annals of Combinatorics 13 (1) pp.27-52 March, 2009
AMS Subject Classification: 05C45, 05C38
Abstract:
A graph $G$ is hamiltonian connected if there exists a hamiltonian path joining any two distinct nodes of $G$. Two hamiltonian paths $P_1 = \big\langle u_1,\, u_2, \ldots,\, u_{\nu(G)} \big\rangle$ and $P_2 = \big\langle v_1,\, v_2, \ldots,\, v_{\nu(G)} \big\rangle$ of $G$ from $u$ to $v$ are independent if $u = u_1 = v_1$, $v = u_{\nu(G)} = v_{\nu(G)}$, and $u_i \ne v_i$ for every $1 < i < \nu(G)$. A set of hamiltonian paths, $\{ P_1,\, P_2, \ldots,\, P_k \}$, of $G$ from $u$ to $v$ are mutually independent if any two different hamiltonian paths are independent from $u$ to $v$. A graph is $k$ mutually independent hamiltonian connected if for any two distinct nodes $u$ and $v$, there are $k$ mutually independent hamiltonian paths from $u$ to $v$. The mutually independent hamiltonian connectivity of a graph $G$, $IHP(G)$, is the maximum integer $k$ such that $G$ is $k$ mutually independent hamiltonian connected. Let $n$ and $k$ be any two distinct positive integers with $n - k \ge 2$. We use $S_{n,\,k}$ to denote the $(n,\,k)$-star graph. In this paper, we prove that $IHP(S_{n,\,k}) = n - 2$ except for $S_{4,\,2}$ such that $IHP(S_{4,\,2}) = 1$.
Keywords: hamiltonian, hamiltonian connected, (n,k)-star graphs

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