- Definition Properties
- Example Hamiltonian decompositions
- Notes
- References
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 
and 
be two graphs with the same number of edges where has more vertices than . Then we say that is an amalgamation of if there is a bijection 
and a surjection 
and the following hold:
- If
, 
are two vertices in where 
, and both and are adjacent by edge 
in , then 
and 
are adjacent by edge 
in . - If is a loop on a vertex
, then is a loop on 
. - If joins
, where 
, but 
, then is a loop on .[3]
Note that while can be a graph or a pseudograph, it will usually be the case that is a pseudograph.
Properties
Edge colorings are invariant to amalgamation. This is obvious, as all of the edges between the two graphs are in bijection with each other. However, what may not be obvious, is that if is a complete graph of the form 
, and we color the edges as to specify a Hamiltonian decomposition (a decomposition into Hamiltonian paths, then those edges also form a Hamiltonian Decomposition in .
Example
Figure 1 illustrates an amalgamation of 
. The invariance of edge coloring and Hamiltonian Decomposition can be seen clearly. The function 
is a bijection and is given as letters in the figure. The function 
is given in the table below.
 |  |
|---|---|
 |  |
 | |
 |  |
 |  |
 | |
Hamiltonian decompositions
One of the ways in which amalgamations can be used is to find Hamiltonian Decompositions of complete graphs with 2n + 1 vertices.[4] The idea is to take a graph and produce an amalgamation of it which is edge colored in 
colors and satisfies certain properties (called an outline Hamiltonian decomposition). We can then 'reverse' the amalgamation and we are left with colored in a Hamiltonian Decomposition.
In [3] Hilton outlines a method for doing this, as well as a method for finding all Hamiltonian Decompositions without repetition. The methods rely on a theorem he provides which states (roughly) that if we have an outline Hamiltonian decomposition, we could have arrived at it by first starting with a Hamiltonian decomposition of the complete graph and then finding an amalgamation for it.
Notes
2. ^Gross 2011
3. ^1 Hilton 1984
4. ^Bahmanian, Amin; Rodger, Chris 2012
References
- Bahmanian, Amin; Rodger, Chris (2012), "What Are Graph Amalgamations?", Auburn University
- Hilton, A. J. W (1984), [https://www.sciencedirect.com/science/article/pii/0095895684900200 "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
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