Isomorphic components of Kronecker product of bipartite graphs

Jha, Pranava K.; Klavžar, Sandi; Zmazek, Blaž

doi:10.7151/dmgt.1057

Cited by 24 publications

(10 citation statements)

References 10 publications

(8 reference statements)

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“…Otherwise, G consists of 2 k−1 isomorphic connected components, where k is the number of i 's that are even, cf. [10]. Since a direct product of graphs is vertex transitive if and only if every factor is vertex transitive, cf.…”

Section: Preliminariesmentioning

confidence: 99%

An almost complete description of perfect codes in direct products of cycles

Klavžar

Špacapan

Žerovnik

2006

Advances in Applied Mathematics

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Let G = × n i=1 C i be a direct product of cycles. It is proved that for any r 1, and any n 2, each connected component of G contains an r-perfect code provided that each i is a multiple of r n + (r + 1) n . On the other hand, if a code of G contains a given vertex and its canonical local vertices, then any i is a multiple of r n + (r + 1) n . It is also proved that an r-perfect code (r 2) of G is uniquely determined by n vertices, and it is conjectured that for r 2 no other codes in G exist other than the constructed ones.

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Section: Preliminariesmentioning

confidence: 99%

An almost complete description of perfect codes in direct products of cycles

Klavžar

Špacapan

Žerovnik

2006

Advances in Applied Mathematics

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“…Lemma 3 [21] If G and H are bipartite graphs one of which has property π , then the two components of G × H are isomorphic.…”

Section: Corollary 1 Let G and H Be Non-complete Non-bipartite Connecmentioning

confidence: 99%

Rainbow vertex-connection and graph products

Mao

Yanling

Wang

et al. 2015

International Journal of Computer Mathematics

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A vertex-coloured graph G is said to be rainbow vertex-connected if every two vertices of G are connected by a path whose internal vertices have distinct colours, such a path is called a rainbow path. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colours that are needed in order to make G rainbow vertex-connected. In this paper, we study the rainbow vertexconnection number on the lexicographical, strong, Cartesian and direct product and present several upper bounds for these products of graphs. The rainbow vertex-connection number of some product networks is also investigated in this paper.

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“…A bipartite graph G = (V 0 ∪ V 1 , E) is said to have a property π if G admits of an automorphism ψ such that x ∈ V 0 if and only if ψ(x) ∈ V 1 . For more details, we refer to [23].…”

Section: The Direct Productmentioning

confidence: 99%

Proper Connection Number of Graph Products

Mao

Yanling

Wang

et al. 2017

Bull. Malays. Math. Sci. Soc.

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A path P in an edge-colored graph G is called a proper path if no two adjacent edges of P are colored the same, and G is proper connected if every two vertices of G are connected by a proper path in G. The proper connection number of a connected graph G, denoted by pc(G), is the minimum number of colors that are needed to make G proper connected. In this paper, we study the proper connection number on the lexicographical, strong, Cartesian, and direct product and present several upper bounds for these products of graphs.

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Isomorphic components of Kronecker product of bipartite graphs

Cited by 24 publications

References 10 publications

An almost complete description of perfect codes in direct products of cycles

An almost complete description of perfect codes in direct products of cycles

Rainbow vertex-connection and graph products

Proper Connection Number of Graph Products

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