a b s t r a c tA k-container of G between u and v, C (u, v), is a set of k internally disjoint paths between u and v. A k * -container C (u, v) of G is a k-container if it contains all vertices of G.A graph G is k * -connected if there exists a k * -container between any two distinct vertices. Thus, every 1 * -connected graph is Hamiltonian connected. Moreover, every 2 * -connected graph is Hamiltonian. Zhan proved that G = L(M) is Hamiltonian connected if the edgeconnectivity of M is at least 4. In this paper, we generalize this result by proving G = L(M)is k * -connected if the edge-connectivity of M is at least max{2k, 4}. We also generalize our result into spanning fan-connectivity.
A graph G is k-ordered if for any sequence of k distinct vertices of G, there exists a cycle in G containing these k vertices in the specified order. It is k-ordered Hamiltonian if, in addition, the required cycle is Hamiltonian. The question of the existence of an infinite class of 3-regular 4-ordered Hamiltonian graphs was posed in 1997 by Ng and Schultz.13At the time, the only known examples were K4and K3,3. Some progress was made in 2008 by Mészáros,12when the Peterson graph was found to be 4-ordered and the Heawood graph was proved to be 4-ordered Hamiltonian; moreover, an infinite class of 3-regular 4-ordered graphs was found. In 2010 a subclass of generalized Petersen graphs was shown to be 4-ordered by Hsu et al.,10with an infinite subset of this subclass being 4-ordered Hamiltonian, thus answering the open question. In this paper we find another infinite class of 3-regular 4-ordered Hamiltonian graphs, part of a subclass of the chordal ring graphs. In addition, we classify precisely which of these graphs are 4-ordered Hamiltonian.
A fe-regular hamiltonian and hamiltonian connected graph G is optimal fault-tolerant hamiltonian and hamiltonian connected if G remains hamiltonian after removing at most k-2 nodes and/or edges and remains hamiltonian connected after removing at most fe -3 nodes and/or edges. In this paper, we investigate a construction scheme to construct optimal fault-tolerant hamiltonian and hamiltonian connected graphs. Hence, some of the generalized hypercubes. Twisted-cubes, Crossed-cubes, and Mobius cubes are optimal fault-tolerant hamiltonian and optimal fault-tolerant hamiltonian connected.
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