# Core (graph theory)

In the mathematical field of graph theory, a core is a notion that describes behavior of a graph with respect to graph homomorphisms.

## Definition

Graph $G$ is a core if every homomorphism $f:G \to G$ is an isomorphism, that is it is a bijection of vertices of $G$.

A core of a graph $H$ is a graph $G$ such that

1. There exists a homomorphism from $H$ to $G$,
2. there exists a homomorphism from $G$ to $H$, and
3. $G$ is minimal with this property.

Two graphs are said to be homomorphism equivalent or hom-equivalent if they have isomorphic cores.

## Examples

• Any complete graph is a core.
• A cycle of odd length is its own core.
• Every two cycles of even length, and more generally every two bipartite graphs are hom-equivalent. The core of each of these graphs is the two-vertex complete graph K2.

## Properties

Every graph has a core, which is determined uniquely, up to isomorphism. The core of a graph G is always an induced subgraph of G. If $G \to H$ and $H \to G$ then the graphs $G$ and $H$ are necessarily hom-equivalent.