Say a simple graph $G$, which is not necessarily planar, has a circumference of $n$ (that is to say, there exists a subgraph $C_n$ where $n$ is maximized). Is it sufficient to say that the graph can then be $n$-colored?
I feel like if $n=3$, then this holds true. I cannot fathom a graph whose largest cyclic subgraph is $C_3$ that's not planar, and furthermore, I feel like it must be some combination of path graphs and copies of $C_3$. In fact, for all planar graphs where $n\geq 4$, this is a trivial matter. What I'm more concerned about is when we're dealing with a non-planar graph with cicumference $n$.
As a final note, I know that for a graph $H$ with genus $1$, the maximum number of colors needed to color $H$ is $7$, so I suppose a good place to start would be to ask, is there a graph of genus $1$ and circumference $6$ that can only be $7$-colored?