The chromatic number of a graph as a functor
Sure, category theorists know such things, but I think for the difficult results that graph theorists care about, the generalities of category theory may at best help as an organizing principle. This is a bit similar to the theory of combinatorial species, which is a way of doing combinatorics inspired by category theory.

Here are some other basic observations, you can read more about such things in Lawvere's Qualitative distinctions between some toposes of generalized graphs, Categories in computer science and logic (Boulder, CO, 1987), volume 92 of Contemporary Mathematics, 261–299. American Mathematical Society.

The category of directed graphs is a presheaf topos on the site consisting of two parallel arrows (so that a presheaf is given by two sets $E$ and $V$ and maps $t : E \to V$ and $s : E \to V$), see A Guided Tour in the Topos of Graphs by Sebastiano Vigna.

For any given $n$, the category of $n$-colored graphs is a topos, because it is just the slice of the topos of graphs over the complete graph on $n$ vertices (an $n$-coloring of $G$ is just a morphism $G \to K_n$).

As Lawvere explains, several other categories of graphs are toposes because they are (equivalent to) toposes of monoid actions for a suitably chosen $M$. Let's just do one. Consider the monoid $M$ of endomaps on $2 = \{0,1\}$. There are four maps $2 \to 2$, we give them names: the identity map $i$, the constant map $0$, the constant map $1$, and the twist map $t$.

An object in the topos of $M$-sets is a set $S$ with a right $M$ action, which is a map $s : S \times M \to S$ satisfying $s(x, i) = x$ and $s(x, f \circ g) = s(s(x, f), g)$. Since there are only four elements in $M$, we might as well figure out what this means. Let $S_0 = \{x \in S \mid \exists y \in S \,.\, x = s(y, 0)\}$ be the set of those elements of $S$ that arise as actions of~$0$. We define $S_1$ similarly. Because $0 \circ g = 0$, it follows for every $g \in M$ and $x \in S_0$ that
$$s(x, g) = s(s(y,0), g) = s(y, 0 \circ g) = s(y, 0) = x.$$
So, the elements of $S_0$ are fixed by the action. Similarly, the elements of $S_1$ are fixed by the action.

To every $M$-set $(S,s)$ we associate a graph $G_S$ as follows. The vertices of $G_S$ are the elements of $S_0 \cup S_1$. The half-edges of $G_S$ are the elements of $S$. Given a half-edge $e \in S$, its source is the vertex $s(e, 0)$. The opposite half-edge of $e$ is $S(e, t)$, and because $t \circ t = 1$, tit follows that the opposite of the opposite is the original half-edge. An edge is a half-edge together with its opposite. The target of a half-edge is the source of its opposite half-edge, i.e., the target of $e$ is $s(s(e,t),0) = s(e,1)$. (There is an anomaly which we allow: a half-edge may be its own opposite.)

What sort of graphs did we get? They are symmetric graphs in which each edge consists of two half-edges. An edge may be degenerate when it is composed of two copies of the same half-edge. In addition, the graphs are reflexive in the sense that each vertex has a distinguished degenerate loop starting and ending at the vertex (this is because $S_0 \cup S_1 \subseteq S$).

Exercise: figure out what the morphisms are, and then show that the description in terms of reflexive symmetric graphs gives a category that it equivalent to the topos of $M$-sets.

Is this the sort of thing you were asking about?