Let $C$ be an odd cycle with vertices $\{v\_1,\dots,v\_n\}$. Let $U = \{u\_1,\dots,u\_m\}$ be another disjoint set of vertices and $m \le n$. Each vertex $v\_i$ of $C$ is adjacent to exactly $1$ vertex in $U$, and each vertex $u\_i$ in $U$ is adjacent to at least $1$ vertex $v\_j$ of $C$. 

Suppose that $U$ has a fixed $2$-coloring with exactly $2$ colors: $\{1, 2\}$. My guess is that $C$ can be colored with 3 colors $\{1,2,3\}$ so that the coloring of the whole graph is valid. I've done some examples, but I haven't been able to see if or why it's true in general.

This question came up when I was looking at algorithmic reduction of hypergraph 2-colorability to graph 3-colorability. 

What I've done in my examples: let the vertices of $C$ be in that order of traversal. For each $v\_i$ in order, choose the smallest color available. The next vertex $v\_{i+1}$ has at most 2 constraints: from $v\_i$ and some $u\_j\in U$. So choose the smallest available color for $v\_{j+1}$. Continue this way. The problem is the last vertex $v\_n$.