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An edge colouring assigns a label called a colour to every edge of a graph: it is a function $c:E(G)\to \{1,\dots ,k\}$. Here this definition allows arbitrary assignments. Two edges sharing a vertex may receive the same colour. A proper edge colouring imposes the additional condition that such edges receive different colours. Some textbooks use "edge colouring" as shorthand for this proper version; check the convention before applying a theorem.^[ref]

The distinction matters in Ramsey theory. In a red-blue edge colouring of a complete graph, a triangle in a graph is monochromatic when its three edges have the same colour. These are arbitrary edge colourings: forbidding equal colours on edges sharing a vertex would forbid the very monochromatic triangles in a graph being sought. For example, assigning red to every edge of $K_3$ is a valid arbitrary edge colouring, although it is not a proper edge colouring. Colour labels can also stand for time slots: then proper edge colouring turns incompatible jobs into separate rounds.

An edge joins two vertices of a graph; in a simple graph it is an unordered pair of distinct vertices.

An edge of a graph joins endpoints in a graph. For a simple graph it is an unordered pair of distinct vertices of a graph; for a directed graph its endpoints are ordered.

An edge of a hypergraph is one of its specified subsets of the vertex set. In a uniform hypergraph all edges have the same cardinality; unlike an ordinary edge of a graph, an edge may contain more than two vertices.

A functional graph has exactly one outgoing edge of a graph at every vertex of a graph, representing a function from its vertex of a graph set to itself. In a finite functional graph, each weakly connected component contains one directed cycle, with directed trees feeding into it. Iterating the represented function eventually reaches that directed cycle.

A hyperedge is a member of the edge family of a hypergraph. In an $r$-uniform hypergraph it is an $r$-element subset of the vertex set. For $r=2$ it is an ordinary edge of a graph. A simple hypergraph has no repeated hyperedges; a multihypergraph records multiplicities.

An image edge is a rapid grey-value transition, modeled as a large gradient or a jump discontinuity. It differs from an edge of a graph. Variational models can penalize image edge length, while diffusion image processing estimates image edge directions to limit image smoothing across them.

The binomial random graph $G(n,p)$ has vertex set $[n]=\{1,\dots ,n\}$, and each unordered pair of distinct vertices is included as an edge independently with probability $p$.

Let $X_t$ count copies of the complete graph $K_t$. Each $t$-element vertex set forms a copy with probability $p^{\left(\genfrac{}{}{0ex}{}{t}{2}\right)}$, so linearity of expectation givesThe event $E_t$ is $\{X_t\ge 1\}$. By the first moment method,This is the clique count in a binomial random graph.

With an adjacency list, initialization and the outer loop cost $O(|V|)$. Each vertex is visited once, and the adjacency scans inspect each edge of a graph twice, once from each endpoint. The total worst-case time is therefore $O(|V|+|E|)$, with $O(|V|)$ labels and recursion-stack space. An adjacency matrix representation instead incurs $O(|V{|}^2)$ scanning time.