We have come up with a sequence of generation theorems which each build upon the last class. Whilst this has enabled us to obtain recursive structures for multi- ple subclasses of quartic planar graphs in one fell swoop, it is quite possible that some of the later classes can be generated using fewer operations. We already know that this is the case, for instance, with arbitrary quadrangulations.
3.4. FURTHER WORK 43
Away from quartic planar graphs and quadrangulations of the sphere, we see two broad directions in which to take generating theorems for maps. The first is to look to generate restricted subclasses of pentangulations and 5-regular plane graphs – there are good results for classes of cubic and quartic graphs that satisfy certain connectedness or degree conditions, but still very few for the 5- regular case. This is likely to be difficult, however, given the complexity of the operations required in [43].
The second possibility is to work on higher-genus surfaces. The proofs of the theorems we mentioned before for the characterisation of irreducible graphs were all particular to the given surface, and so do not generalise well. This was also the case for early characterisations of irreducible triangulations, until Sulanke considered genus-increasing expansion operations that relate the irreducible tri- angulations on different surfaces [78, 81]. There are two basic operations: given a triangle, the equivalent to adding a handle to the surface is given by removing two faces and then identifying their boundaries, and similarly, a crosscap can be added by removing a vertex of degree six and identifying each neighbour to its opposite. In both cases, if the faces and vertices involved are sufficiently far apart, then this produdces a triangulation of the original surface with a handle or crosscap added.
The generation theorem implied by this has been implemented to determine complete lists of irreducible triangulations on orientable surfaces up to genus 2 and non-orientable surfaces up to genus 4. One can then generate all quadrangulations of these surfaces by some edges in all possible ways. On the other hand, by defining analogous handle- and crosscap-addition expansions using quadrangular faces, we believe it may be possible to also adapt Sulanke’s strategy to generate quadrangulations directly.
Chapter 4
Self-avoiding Eulerian circuits
An Eulerian circuit of a graph is a circuit in which every edge of the graph is traversed exactly once. Any graph that admits such a circuit is said to be
Eulerian. It is a standard fact, attributed to Euler∗, that a simple graph is Eulerian if and only if it is connected and all of its vertices have even degree. If one is asked to find an Eulerian circuit given some Eulerian graph, one can breathe a sigh of relief because this is very easy to do. Here are two well-known algorithms.
1. (Fleury) Starting at an arbitrary vertex, choose a sequence of unused edges such that at each step the subgraph induced by the set of unused vertices is connected. It is straightforward to prove that the resulting circuit uses all edges, with the basic principle being that even degrees ensure that when the circuit enters any vertex except the initial one, there is always an edge available for it to exit.
2. (Hierholzer) Again starting at an arbitrary vertex, choose any sequence of available edges until we form a circuitC. If there are any edges of the graph that are not in the circuit, then we pick a vertex v on C that is adjacent to an unused edge and choose another sequence of available edges until we return to v, thus creating another circuit D. These circuits can be joined by following one vw-subpath in C, then taking D, and finally taking the other wv-subpath in C to return to the start. This is repeated until all edges have been incorporated.
The latter procedure highlights an interesting feature of Eulerian circuits that
∗For a fun historical diversion, we recommend [75]. In particular, Euler proved necessity in
1735, but a proof of sufficiency was not published until 1873 by Hierholzer.
46 CHAPTER 4. SELF-AVOIDING EULERIAN CIRCUITS
is present but somewhat obscured in the first approach; provided our graph has vertices of degree greater than 2, any Eulerian circuit will cross back on itself many times and hence contain subcircuits and subcycles. More precisely, we make the following definition.
Definition 4.1. LetC =x1x2. . . xn be an Eulerian circuit in a simple graph. A
subcycle of C is a subpath xixi+1. . . xj of C such that xi = xj and the vertices
xi, xi+1, . . . , xj−1 are all distinct.
A subcircuit of an Eulerian circuit is defined by dropping the condition that the internal vertices are distinct, but since any subcircuit contains a subcycle, we find it unnecessary to use both terms. This raises the question of whether we can control the lengths of these subcycles for particular classes of graphs. Provided the girth of the graph is small enough, it is generally straightforward to find an Eulerian circuit with a small subcycle, so we are interested in when it is possible to find one that avoids small subcycles.
From a graph theoretic perspective, the study of these restricted Eulerian circuits is partly motivated by their connection to tree, path and cycle decompo- sitions of graphs surrounding which there are numerous famous open problems. We will defer this discussion to the next section. In addition, Eulerian circuits have found applications in other areas such as in bioinformatics, where the prob- lem of assembling DNA fragments after sequencing can in some instances be reduced to finding an Eulerian circuit of a particular auxiliary graph [70]. With a view to maximising the potential for these applications, it is desirable for our proofs to be constructive.