It is known that for graphs $A$ and $B$ with odd cycles, the direct product $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. But this is not necessarily true if one of $A$ or $B$ is bipartite, and until now there has been no characterization of such vertex-transitive direct products. We prove that if $A$ and $B$ are both bipartite, or both non-bipartite, then $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. Also, if $A$ has an odd cycle and $B$ is bipartite, then $A\times B$ is vertex-transitive if and only if both $A\times K_2$ and $B$ are vertex-transitive.
@article{10_37236_6999,
author = {Richard H. Hammack and Wilfried Imrich},
title = {Vertex-transitive direct products of graphs},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {2},
doi = {10.37236/6999},
zbl = {1391.05217},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6999/}
}
TY - JOUR
AU - Richard H. Hammack
AU - Wilfried Imrich
TI - Vertex-transitive direct products of graphs
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/6999/
DO - 10.37236/6999
ID - 10_37236_6999
ER -
%0 Journal Article
%A Richard H. Hammack
%A Wilfried Imrich
%T Vertex-transitive direct products of graphs
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/6999/
%R 10.37236/6999
%F 10_37236_6999
Richard H. Hammack; Wilfried Imrich. Vertex-transitive direct products of graphs. The electronic journal of combinatorics, Tome 25 (2018) no. 2. doi: 10.37236/6999
