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2-Tone Colorings in Graph Products

100%

Loe J. , Middelbrooks D. , Morris A. , Wash K.

tom 35

nr 1

55-72

A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set {1, . . . , k}, where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a t- tone k-coloring is known as the t-tone chromatic number. We study the 2-tone chromatic number in three different graph products. In particular, given graphs G and H, we bound the 2-tone chromatic number for the direct product G×H, the Cartesian product G□H, and the strong product G⊠H.

Harary Index of Product Graphs

100%

Pattabiraman K. , Paulraja P.

tom 35

nr 1

17-33

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, the exact formulae for the Harary indices of tensor product G × Km0,m1,...,mr−1 and the strong product G⊠Km0,m1,...,mr−1 , where Km0,m1,...,mr−1 is the complete multipartite graph with partite sets of sizes m0,m1, . . . ,mr−1 are obtained. Also upper bounds for the Harary indices of tensor and strong products of graphs are estabilished. Finally, the exact formula for the Harary index of the wreath product G ○ G′ is obtained.

Sharp Upper Bounds for Generalized Edge-Connectivity of Product Graphs

100%

Sun Y.

Discussiones Mathematicae Graph Theory

2016

tom 36

nr 4

833-843

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.

On the independence number of some strong products of cycle-powers

100%

Jurkiewicz M. , Kubale M. , Ocetkiewicz K.

Foundations of Computing and Decision Sciences

2015

tom Vol. 40, No. 2

133--141

In the paper we give some theoretical and computational results on the third strong power of cycle-powers, for example, we have found the independence numbers α((C210)⊠3) = 30 and α((C414)⊠3) = 14. A number of optimizations have been introduced to improve the running time of our exhaustive algorithm used to establish the independence number of the third strong power of cycle-powers. Moreover, our results establish new exact values and/or lower bounds on the Shannon capacity of noisy channels.