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2 Discussiones Mathematicae Graph Theory

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Edge-Transitive Lexicographic and Cartesian Products

100%

Imrich W. , Iranmanesh A. , Klavžar S. , Soltani A.

Discussiones Mathematicae Graph Theory

2016

tom 36

nr 4

857-865

In this note connected, edge-transitive lexicographic and Cartesian products are characterized. For the lexicographic product G ◦ H of a connected graph G that is not complete by a graph H, we show that it is edge-transitive if and only if G is edge-transitive and H is edgeless. If the first factor of G ∘ H is non-trivial and complete, then G ∘ H is edge-transitive if and only if H is the lexicographic product of a complete graph by an edgeless graph. This fixes an error of Li, Wang, Xu, and Zhao [11]. For the Cartesian product it is shown that every connected Cartesian product of at least two non-trivial factors is edge-transitive if and only if it is the Cartesian power of a connected, edge- and vertex-transitive graph.

Distinguishing Cartesian Products of Countable Graphs

88%

Estaji E. , Imrich W. , Kalinowski R. , Pilśniak M. , Tucker T.

Discussiones Mathematicae Graph Theory

2017

tom 37

nr 1

155-164

The distinguishing number D(G) of a graph G is the minimum number of colors needed to color the vertices of G such that the coloring is preserved only by the trivial automorphism. In this paper we improve results about the distinguishing number of Cartesian products of finite and infinite graphs by removing restrictions to prime or relatively prime factors.