A graph homomorphism from a graph to a graph , written , is a mapping from the vertex set of to the vertex set of such that implies .
The above definition is extended to directed graphs. Then, for a homomorphism , is an arc of if is an arc of .
If there exists a homomorphism we shall write , and otherwise. If , is said to be homomorphic to or -colourable.
Two graphs and are homomorphically equivalent if and .
A retract of a graph is a subgraph of such that there exists a homomorphism , called retraction with for any vertex of . A core is a graph which does not retract to a proper subgraph. Any graph is homomorphically equivalent to a unique core.
The composition of homomorphisms are homomorphisms.
Graph homomorphism preserves connectedness.
Connection to coloring and girth
A graph coloring is an assignment of one of k colors to a graph G so that the endpoints of each edge have different colors, for some number k. Any coloring corresponds to a homomorphism from G to a complete graph Kk: the vertices of Kk correspond to the colors of G, and f maps each vertex of G with color c to the vertex of Kk that corresponds to c. This is a valid homomorphism because the endpoints of each edge of G are mapped to distinct vertices of Kk, and every two distinct vertices of Kk are connected by an edge, so every edge in G is mapped to an adjacent pair of vertices in Kk. Conversely if f is a homomorphism from G to Kk, then one can color G by using the same color for two vertices in G whenever they are both mapped to the same vertex in Kk. Because Kk has no edges that connect a vertex to itself, it is not possible for two adjacent vertices in G to both be mapped to the same vertex in Kk, so this gives a valid coloring. That is, G has a k-coloring if and only if it has a homomorphism to Kk.
If there are two homomorphisms , then their composition is also a homomorphism. In other words, if a graph G can be colored with k colors, and there is a homomorphism , then H can also be k-colored. Therefore, whenever a homomorphism exists, the chromatic number of H is less than or equal to the chromatic number of G.
Homomorphisms can also be used very similarly to characterize the odd girth of a graph G, the length of its shortest odd-length cycle. The odd girth is, equivalently, the smallest odd number g for which there exists a homomorphism . For this reason, if , then the odd girth of G is greater than or equal to the corresponding invariant of H.
The associated decision problem, i.e. deciding whether there exists a homomorphism from one graph to another, is NP-complete. Determining whether there is an isomorphism between two graphs is also an important problem in computational complexity theory; see graph isomorphism problem.
- Hell & Nešetřil (2004), p. 7.
- Hell, Pavol; Nešetřil, Jaroslav (2004), Graphs and Homomorphisms (Oxford Lecture Series in Mathematics and Its Applications), Oxford University Press, ISBN 0-19-852817-5
- Mitchell, James D. (2011), Endomorphism monoids of all connected graphs with at most 7 vertices.
- R. Brown, I. Morris, J. Shrimpton and C.D. Wensley, `Graphs of Morphisms of Graphs', Electronic Journal of Combinatorics, A1 of Volume 15(1), 2008. 1-28.