Mutually independent Hamiltonian cycles in alternating group graphs

Su, Hsun; Chen, Shih-Yan; Kao, Shin-Shin

doi:10.1007/s11227-011-0614-4

Cited by 10 publications

(3 citation statements)

References 19 publications

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“…Proof. We use the information in [10] to partition the edges of G to n − 2 edge-disjoint Hamiltonian cycles. Then, we bi-color each disjoint Hamiltonian cycle to obtain the class 1 coloring of the edges of G.…”

Section: Cayley Graphs Of Permutation Groups With Transposition Gener...mentioning

confidence: 99%

Total Coloring of Some Classes of Cayley Graphs on Non-Abelian Groups

et al. 2022

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Total Coloring of a graph G is a type of graph coloring in which any two adjacent vertices, an edge, and its incident vertices or any two adjacent edges do not receive the same color. The minimum number of colors required for the total coloring of a graph is called the total chromatic number of the graph, denoted by χ″(G). Mehdi Behzad and Vadim Vizing simultaneously worked on the total colorings and proposed the Total Coloring Conjecture (TCC). The conjecture states that the maximum number of colors required in a total coloring is Δ(G)+2, where Δ(G) is the maximum degree of the graph G. Graphs derived from the symmetric groups are robust graph structures used in interconnection networks and distributed computing. The TCC is still open for the circulant graphs. In this paper, we derive the upper bounds for χ″(G) of some classes of Cayley graphs on non-abelian groups, typically Cayley graphs on the symmetric groups and dihedral groups. We also obtain the upper bounds of the total chromatic number of complements of Kneser graphs.

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Section: Cayley Graphs Of Permutation Groups With Transposition Gener...mentioning

confidence: 99%

Total Coloring of Some Classes of Cayley Graphs on Non-Abelian Groups

et al. 2022

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“…For the two families of graphs, many researchers were attracted to study fault tolerant routing [12], fault tolerant embedding [5], [6], [42], matching preclusion [2], [11], restricted connectivity [15], [25], [35], [36], [48] and diagnosability [10], [25], [30], [34]- [36], [41]. Moreover, alternating group graphs are also edge-transitive and possess stronger and rich properties on Hamiltonicity (e.g., it has been shown to be not only pancyclic and Hamiltonian-connected [33] but also panconnected [6], panpositionable [40] and mutually independent Hamiltonian [39]). The following structural property disclosed by Cheng et al [18] is of particular interest and closely related to -component connectivity.…”

Section: B Literature Related To Alternating Group Graph and Split-starsmentioning

confidence: 99%

Measuring the Vulnerability of Alternating Group Graphs and Split-Star Networks in Terms of Component Connectivity

Hao

Chang

2019

IEEE Access

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For an integer 2, the -component connectivity of a graph G, denoted by κ (G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least components or a graph with fewer than vertices. This is a natural generalization of the classical connectivity of graphs defined in term of the minimum vertex-cut and a good measure of vulnerability for the graph corresponding to a network. So far, the exact values of -connectivity are known only for a few classes of networks and small 's. It has been pointed out in component connectivity of the hypercubes, International Journal of Computer Mathematics 89 (2012) 137-145] that determining -connectivity is still unsolved for most interconnection networks such as alternating group graphs and star graphs. In this paper, by exploring the combinatorial properties and the fault-tolerance of the alternating group graphs AG n and a variation of the star graphs called split-stars S 2 n , we study their -component connectivities. We obtain the following results: 1) κ 3 (AG n ) = 4n − 10 and κ 4 (AG n ) = 6n − 16 for n 4, and κ 5 (AG n ) = 8n − 24 for n 5 and 2) κ 3 (S 2 n ) = 4n − 8, κ 4 (S 2 n ) = 6n − 14, and κ 5 (S 2 n ) = 8n − 20 for n 4.

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“…For the two families of graphs, many researchers were attracted to study fault tolerant routing [10], fault tolerant embedding [5,6,37], matching preclusion [2,9], restricted connectivity [13,22,30,31,39] and diagnosability [8,22,25,[29][30][31]36]. Moreover, alternating group graphs are also edge-transitive and possess stronger and rich properties on Hamiltonicity (e.g., it has been shown to be not only pancyclic and Hamiltonian-connected [28] but also panconnected [6], panpositionable [35] and mutually independent Hamiltonian [34]). The following structural property disclosed by Cheng et al [16] is of particular interest and closely related to -component connectivity.…”

Section: Introductionmentioning

confidence: 99%

The Component Connectivity of Alternating Group Graphs and Split-Stars

Gu,

Hao,

Chang

2018

Preprint

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For an integer 2, the -component connectivity of a graph G, denoted by κ (G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least components or a graph with fewer than vertices. This is a natural generalization of the classical connectivity of graphs defined in term of the minimum vertex-cut and is a good measure of robustness for the graph corresponding to a network. So far, the exact values of -connectivity are known only for a few classes of networks and small 's. It has been pointed out in [Component connectivity of the hypercubes, Int. J. Comput. Math. 89 (2012) 137-145] that determining -connectivity is still unsolved for most interconnection networks, such as alternating group graphs and star graphs. In this paper, by exploring the combinatorial properties and fault-tolerance of the alternating group graphs AG n and a variation of the star graphs called split-stars S 2 n , we study their -component connectivities. We obtain the following results: (i) κ 3 (AG n ) = 4n−10 and κ 4 (AG n ) = 6n−16 for n 4, and κ 5 (AG n ) = 8n − 24 for n 5; (ii) κ 3 (S 2 n ) = 4n − 8, κ 4 (S 2 n ) = 6n − 14, and κ 5 (S 2 n ) = 8n − 20 for n 4.

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Mutually independent Hamiltonian cycles in alternating group graphs

Cited by 10 publications

References 19 publications

Total Coloring of Some Classes of Cayley Graphs on Non-Abelian Groups

Total Coloring of Some Classes of Cayley Graphs on Non-Abelian Groups

Measuring the Vulnerability of Alternating Group Graphs and Split-Star Networks in Terms of Component Connectivity

The Component Connectivity of Alternating Group Graphs and Split-Stars

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