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Is this right?

Let G1 and G2 be two regular graphs. If the Kronecker product of the complete graph K2 and G1 is isomorphic to the Kronecker product of the complete graph K2 and G2, then G1 is isomorphic to G2?

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No. W. Imrich and T. Pisanski proved (in Multiple Kronecker covering graphs, Eur. J. Combin. 29 (2008), 1116-1122) that the Desargues graph can be represented as the Kronecker covering graph of two non-isomorphic cubic graphs. (The recent paper link would also be helpful!)

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  • $\begingroup$ Thank you very much Yusuf! $\endgroup$ Commented Dec 27, 2022 at 21:40
  • $\begingroup$ You're welcome, and hope that helps. $\endgroup$ Commented Dec 28, 2022 at 7:55
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Also the bipartite double covers of the Shrikhande graph and the 4x4 Rook graph (i.e., the cartesian product of $K_4$ with itself) are isomorphic. Both of these graphs are 6-regular.

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