The chromatic index of a graph. This is the minimal number of colours needed such that no two adjacent edges have the same colour.

The Alon-Tarsi number of a graph. Let $G$ be a graph with vertices $\{1,\dots,n\}$ and edge set $E$. Let $P_G=\prod_{i < j, (i,j)\in E} x_i-x_j$ be its graph polynomial. Then the Alon-Tarsi number is the smallest number $k$ such that $P_G$ contains a monomial with exponents strictly less than $k$.

The 2-dynamic chromatic number of a graph. A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $2$-dynamic chromatic number of a graph.

The acyclic chromatic number of a graph. This is the smallest size of a vertex partition $\{V_1,\dots,V_k\}$ such that each $V_i$ is an independent set and for all $i,j$ the subgraph inducted by $V_i\cup V_j$ does not contain a cycle.

The size of the core of a graph. The core of the graph $G$ is the smallest graph $C$ such that there is a graph homomorphism from $G$ to $C$ and a graph homomorphism from $C$ to $G$.

The game chromatic number of a graph. Two players, Alice and Bob, take turns colouring properly any uncolored vertex of the graph. Alice begins. If it is not possible for either player to colour a vertex, then Bob wins. If the graph is completely colored, Alice wins. The game chromatic number is the smallest number of colours such that Alice has a winning strategy.