Why is the tensor product of two graphs $G$ and $H$ each containing an odd circle must also contain an odd circle?
The tensor product of two graph $G$ and $H$ has the vertex set the cartesian product of the vertex set of $G$ and $H$ and there is an edge between $(u,v),(x,y)$ if $(u,x) and (v,y)$ are edges in $G$ and $H$ respectively.
I was trying to see for $G=C_3$ and $H=C_5$ but couldn't get an circle for their tensor product.
Let $(X_1, X_2,\ldots X_n)$ and $(Y_1,Y_2,\ldots,Y_m)$ be those cycles ($n, m$ - odd numbers). If you make the cycle $((X_1,Y_1),(X_2,Y_2),\ldots)$ (wrapping around on the first or on the second component as necessary), this cycle will have the length $\text{lcm}(m,n)$. (lcm = Least Common Multiplier.) What remains is to note that lcm of any two odd numbers is again an odd number.