Chromatic number k+2, cycle length 2 mod k Let $G$ be a graph with $\chi(G)=k+2$ for $k\ge3$. Prove that $G$ contains a cycle of a length $l$ such that $l \equiv 2 \;(\bmod\; k)$. 
Not quite sure how to approach this at all. I know that there is a subgraph $H$ with $\delta(H)\ge k+1$ (critical subgraph/degeneracy condition), and therefore $H$ has a cycle with length at least $k+2$, but I don't know how to get the $l \equiv 2 \;(\bmod\; k)$.
Thanks in advance.
 A: I was able to expand my initial thinking about the critical graph into a solution: 
We can assume without loss of generality that $G$ is $(k+2)$-critical (because we can find it as a subgraph). $G$ has $\delta(G)=k+1$. Let $P$ be a maximal (by length) path in $G$ and let $v$ be one of the endpoints. Then $v$ has $k+1$ neighbours and they are all in $P$ (otherwise $P$ is not maximal). Except for the edge connecting $v$ in the path itself, there are $k$ edges not in the path which go from $v$ to vertices in the path, thereby creating (at least) $k$ different cycles. If their lengths are all different mod k then one has to have length 2 mod k. Otherwise there are two cycles $(v,...,u,v)$ and $(v,...,w,v)$ (where the three dots go along $P$) whose lengths are equal mod k, in which case the cycle that is the "difference" between them $(u,...,w,v,u)$ (where the three dots go along $P$) has length 2 mod k as needed.
A: DISCLAIMER: this is NOT an answer to the main question, but a response to a comment.
The general statement is, that when $\chi(G)>k$, then $G$ has a cycle with length congruent to $1\pmod k$.
The proof is a bit intricate, but the parts I leave out are rather easy to prove.
Step 1: Fix a vertex $v$ of $G$.
Step 2: Choose a spanning tree $T$ of $G$ such that $\Sigma_{u\in V(G)} d_T(u,v)$ is maximized.
Step 3: Prove that every edge of $G$ that is not an edge of $T$ has both its endpoints on a path in $T$
starting at $v$.
Step 4: Color vertex $x$ of $T$ with $d_T(x,v)\pmod k$. This cannot be a proper coloring of $G$ since $\chi(G)>k$,
so there is an edge $e$ of $G$ connecting two vertices of the same color. This edge has both its endpoints
on a $T$-path starting at $v$, and since they have the same color, the length of the path between them
is a multiple of $k$. Together with $e$ this completes a cycle with the desired length.
If you need more detail, indicate which part is causing trouble.
