The dual Coxeter number of a finite Cartan type.

The number of distinct orders of elements in a Weyl group of finite Cartan type.

The Grundy number of a graph. The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring. In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).

The burning number of a graph. This is the minimum number of rounds needed to burn all vertices of a graph. In each round, the neighbours of all burned vertices are burnt. Additionally, an unburned vertex may be chosen to be burned.

The open packing number of a graph. This is the size of a largest subset of vertices of a graph, such that any two distinct vertices in the subset have disjoint open neighbourhood.

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.

The Castelnuovo-Mumford regularity of a graph.

The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.

The connected partition number of a graph. This is the maximal number of blocks of a set partition $P$ of the set of vertices of a graph such that contracting each block of $P$ to a single vertex yields a clique. Also called the pseudoachromatic number of a graph. This is the largest $n$ such that there exists a (not necessarily proper) $n$-coloring of the graph so that every two distinct colors are adjacent.

The lettericity of a graph. Let $D$ be a digraph on $k$ vertices, possibly with loops and let $w$ be a word of length $n$ whose letters are vertices of $D$. The letter graph corresponding to $D$ and $w$ is the graph with vertex set $\{1,\dots,n\}$ whose edges are the pairs $(i,j)$ with $i < j$ sucht that $(w_i, w_j)$ is a (directed) edge of $D$.