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.
@article{bwmeta1.element.doi-10_7151_dmgt_1892,
author = {Wilfried Imrich and Ali Iranmanesh and Sandi Klav\v zar and Abolghasem Soltani},
title = {Edge-Transitive Lexicographic and Cartesian Products},
journal = {Discussiones Mathematicae Graph Theory},
volume = {36},
year = {2016},
pages = {857-865},
zbl = {1350.05145},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1892}
}
Wilfried Imrich; Ali Iranmanesh; Sandi Klavžar; Abolghasem Soltani. Edge-Transitive Lexicographic and Cartesian Products. Discussiones Mathematicae Graph Theory, Tome 36 (2016) pp. 857-865. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1892/