In graph theory, a graph amalgamation is a relationship between two graphs (one graph is an amalgamation of another). Similar relationships include subgraphs and minors. Amalgamations can provide a way to reduce a graph to a simpler graph while keeping certain structure intact. The amalgamation can then be used to study properties of the original graph in an easier to understand context. Applications include embeddings,^[1] computing genus distribution,^[2] and Hamiltonian decompositions.

Definition

Let ${\displaystyle G}$ and ${\displaystyle H}$ be two graphs with the same number of edges where ${\displaystyle G}$ has more vertices than ${\displaystyle H}$. Then we say that ${\displaystyle H}$ is an amalgamation of ${\displaystyle G}$ if there is a bijection ${\displaystyle \phi :E(G)\to E(H)}$ and a surjection ${\displaystyle \psi :V(G)\to V(H)}$ and the following hold:

• If ${\displaystyle x}$, ${\displaystyle y}$ are two vertices in ${\displaystyle G}$ where ${\displaystyle \psi (x)\neq \psi (y)}$, and both ${\displaystyle x}$ and ${\displaystyle y}$ are adjacent by edge ${\displaystyle e}$ in ${\displaystyle G}$, then ${\displaystyle \psi (x)}$ and ${\displaystyle \psi (y)}$ are adjacent by edge ${\displaystyle \phi (e)}$ in ${\displaystyle H}$.
• If ${\displaystyle e}$ is a loop on a vertex ${\displaystyle x\in V(G)}$, then ${\displaystyle \phi (e)}$ is a loop on ${\displaystyle \psi (x)\in H}$.
• If ${\displaystyle e}$ joins ${\displaystyle x,y\in V(G)}$, where ${\displaystyle x\neq y}$, but ${\displaystyle \psi (x)=\psi (y)}$, then ${\displaystyle \phi (e)}$ is a loop on ${\displaystyle \psi (x)}$.^[3]

Note that while ${\displaystyle G}$ can be a graph or a pseudograph, it will usually be the case that ${\displaystyle H}$ is a pseudograph.

Example

Figure 1: An amalgamation of the complete graph on five vertices.

Figure 1 illustrates an amalgamation of ${\displaystyle K_{5}}$. The invariance of edge coloring and Hamiltonian Decomposition can be seen clearly. The function ${\displaystyle \phi }$ is a bijection and is given as letters in the figure. The function ${\displaystyle \psi }$ is given in the table below.

${\displaystyle v\in V(G)}$	${\displaystyle \psi (v)}$
${\displaystyle v_{1}}$	${\displaystyle u_{2}}$
${\displaystyle v_{2}}$	${\displaystyle u_{2}}$
${\displaystyle v_{3}}$	${\displaystyle u_{1}}$
${\displaystyle v_{4}}$	${\displaystyle u_{3}}$
${\displaystyle v_{5}}$	${\displaystyle u_{2}}$

Notes

References

• Bahmanian, Amin; Rodger, Chris (2012), "What Are Graph Amalgamations?", Auburn University
• Hilton, A. J. W (1984), "Hamiltonian Decompositions of Complete Graphs, Journal of Combinatorial Theory, Series B 36, 125–134
• Gross, Jonathan L.; Tucker, Thomas W. (1987), Topological Graph Theory, Courier Dover Publications, 151
• Gross, Jonathan L. (2011), "Genus Distributions of Cubic Outerplanar Graphs", Journal of Graph Algorithms and Applications, Vol. 15, no. 2, pp. 295–316