K3K2

K3K2, typically written as K_{3,2}, is the complete bipartite graph with partitions of sizes 3 and 2. It has five vertices and six edges, connecting every vertex in the 3-vertex set to every vertex in the 2-vertex set.

In this graph, the three vertices on the size-3 side each have degree 2, while the two vertices on the size-2 side each have degree 3. The degree sequence is therefore 3, 3, 2, 2, 2. K_{3,2} is bipartite, meaning it contains no odd cycles, and it is connected.

K_{3,2} is planar, and it can be drawn on a plane without edge crossings. The smallest cycle within the graph has length 4, so the girth is 4. The graph serves as a simple example in discussions of planarity and bipartite graphs.

In terms of symmetry, the automorphism group of K_{3,2} is isomorphic to the direct product S3 × S2, with order 12. This reflects the ability to permute the three vertices on the 3-side and the two vertices on the 2-side independently, while preserving adjacencies.

K_{3,2} is a specific instance of the family of complete bipartite graphs K_{m,n}, and it is a subgraph of the larger K_{3,3}. It appears in introductory graph theory as a compact, nontrivial example illustrating bipartite structure, planarity, and symmetry.

See also: complete bipartite graphs, K_{m,n}, bipartite graphs, planar graphs.