The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

The degeneracy of a graph. The degeneracy of a graph $G$ is the maximum of the minimum degrees of the (vertex induced) subgraphs of $G$.

The largest degree of a regular subgraph of a graph. For $k > 2$, it is an NP-complete problem to determine whether a graph has a $k$-regular subgraph, see [1].

The Hadwiger number of the graph. Also known as clique contraction number, this is the size of the largest complete minor.

The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.

The hook length of the last cell along the main diagonal of an integer partition.