2regular

2-regular, or 2-regular graph, refers to an undirected graph in which every vertex has degree 2. In a simple graph this means that each vertex is adjacent to exactly two edges. Such graphs are a basic item in the family of regular graphs and are automatically Eulerian, since all vertices have even degree and every connected component contains a closed trail.

In the finite case, every connected component of a 2-regular graph is a cycle. Therefore a finite 2-regular graph is a disjoint union of cycle graphs C_n for some n ≥ 3. More generally, a 2-regular graph is a disjoint union of cycles; in finite graphs the cycles are finite.

The graph has as many edges as vertices: the sum of degrees equals 2|V|, and with every vertex contributing 2, this yields |E| = |V| in a finite 2-regular graph.

Examples include the cycle C_3 (a triangle) and C_4 (a square). A graph consisting of C_3 together with C_4, i.e., C_3 ∪ C_4, is also 2-regular.

Connected 2-regular graphs are precisely the cycle graphs C_n with n ≥ 3; more generally, a graph is 2-regular if and only if it is a disjoint union of cycles. This property makes 2-regular graphs a natural object of study in graph decompositions and Eulerian considerations.