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1
Content available remote

Sharp Upper Bounds for Generalized Edge-Connectivity of Product Graphs

100%
Sun Y.
Discussiones Mathematicae Graph Theory
|
2016
|
tom 36
|
nr 4
833-843
EN
The generalized k-connectivity κk(G) of a graph G was introduced by Hager in 1985. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k-edge-connectivity which is defined as λk(G) = min{λ(S) : S ⊆ V (G) and |S| = k}, where λ(S) denote the maximum number ℓ of pairwise edge-disjoint trees T1, T2, . . . , Tℓ in G such that S ⊆ V (Ti) for 1 ≤ i ≤ ℓ. In this paper, we study the generalized edge- connectivity of product graphs and obtain sharp upper bounds for the generalized 3-edge-connectivity of Cartesian product graphs and strong product graphs. Among our results, some special cases are also discussed.
2
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On the Maximum and Minimum Sizes of a Graph with Givenk-Connectivity

100%
Sun Y.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 3
623-632
EN
The concept of k-connectivity κk(G), introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. For an integer k ≥ 2, the k-connectivity of a connected graph G with order n ≥ k is the smallest number of vertices whose removal from G produces a graph with at least k components or a graph with fewer than k vertices. In this paper, we get a sharp upper bound for the size of G with κk(G) = t, where 1 ≤ t ≤ n − k and k ≥ 3; moreover, the unique extremal graph is given. Based on this result, we get the exact values for the maximum size, denoted by g(n, k, t), of a connected graph G with order n and κk(G) = t. We also compute the exact values and bounds for another parameter f(n, k, t) which is defined as the minimum size of a connected graph G with order n and κk(G) = t, where 1 ≤ t ≤ n − k and k ≥ 3.
3
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A Sharp Lower Bound For The Generalized 3-Edge-Connectivity Of Strong Product Graphs

100%
Sun Y.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 4
975-988
EN
The generalized k-connectivity κk(G) of a graph G, mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k-edge-connectivity which is defined as λk(G) = min{λG(S) | S ⊆ V (G) and |S| = k}, where λG(S) denote the maximum number ℓ of pairwise edge-disjoint trees T1, T2, . . . , Tℓ in G such that S ⊆ V (Ti) for 1 ≤ i ≤ ℓ. In this paper we get a sharp lower bound for the generalized 3-edge-connectivity of the strong product of any two connected graphs.
4
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On Two Generalized Connectivities of Graphs

51%
Sun Y., Li F., Jin Z.
Discussiones Mathematicae Graph Theory
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2017
|
tom 38
|
nr 1
245-261
EN
The concept of generalized k-connectivity κk(G), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. The pendant tree-connectivity τk(G) was also introduced by Hager in 1985, which is a specialization of generalized k-connectivity but a generalization of the classical connectivity. Another generalized connectivity of a graph G, named k-connectivity κ′k(G), introduced by Chartrand et al. in 1984, is a generalization of the cut-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of κ′k(G) and τk(G) by showing that for a connected graph G of order n, if κ′k(G) ≠ n − k + 1 where k ≥ 3, then 1 ≤ κ′k(G) − τk(G) ≤ n − k; otherwise, 1 ≤ κ′k(G) ‘− τk(G) ≤ n − k + 1. Moreover, all of these bounds are sharp. We get a sharp upper bound for the 3-connectivity of the Cartesian product of any two connected graphs with orders at least 5. Especially, the exact values for some special cases are determined. Among our results, we also study the pendant tree-connectivity of Cayley graphs on Abelian groups of small degrees and obtain the exact values for τk(G), where G is a cubic or 4-regular Cayley graph on Abelian groups, 3 ≤ k ≤ n.
5
Content available remote

Rainbow Connection Number of Graphs with Diameter 3

51%
Li H., Li X., Sun Y.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
|
nr 1
141-154
EN
A path in an edge-colored graph G is rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer k for which there exists a k-edge-coloring of G such that every pair of distinct vertices of G is connected by a rainbow path. Let f(d) denote the minimum number such that rc(G) ≤ f(d) for each bridgeless graph G with diameter d. In this paper, we shall show that 7 ≤ f(3) ≤ 9.
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