Polynomial algorithms for shortest hamiltonian path and. I think when we have a hamiltonian cycle since each vertex lies in the hamiltonian cycle if we consider one vertex as starting and ending cycle. Let x be a connected cayley graph on an abelian group. Pdf cayley graphs arise naturally in computer science, in the study of. There is no easy theorem like eulers theorem to tell if a graph has hamilton. Construct a graph that has neither an euler nor a hamiltonian circuit.
All of the nonhamiltonian cayley digraphs provided by proposition 20 are generated. The cayley diagram of the generators s in g is a directed graph denoted by cays. It is then proven that every cayley digraph of a dedekind group has a hamiltonian path. Pdf hamiltonian cycles and paths in cayley graphs and digraphs. Two examples of math we use on a regular basis are euler and hamiltonian circuits.
Learning outcomes you should have the ability to do the following after this lesson. So we have n1n22 hamiltonian cycle because we should select 2 edges of n1 edges which linked to this vertex. The procedure has intrinsic advantage of landing on the desired solution in quickest possible time and even in worst case in polynomial time. Again similar to euler, the graph is consider to of the graph 3. Pdf lovasz had posed a question stating whether every connected, vertextransitive graph has a hamilton path in 1969. Cayley graphs of groups and their applications semantic. One exception is the theorem by pak and radoicic pr that every finite group with at least three elements has a generating set of size, such that the corresponding cayley graph is hamiltonian. Hamiltonconnected cayley graphs on hamiltonian groups core. The talk will explain what cayley graphs are, describe.
If either g is a finite abelian group or a semidirect product of a cyclic group of prime order by a finite abelian group of odd order, then every connected cayley graph of g is hamiltoman. Hamilton connected cayley graphs on hamiltonian groups article pdf available in european journal of combinatorics 226. T are isomorphic, but they are distinguished via the labels. Hamiltonian paths and cayley digraphs of algebraic groups. Give two examples of situations that could be modeled by a graph in which finding a hamiltonian path or circuit would be of benefit. Rankin found a simple arithmetic criterion that determines which are hamiltonian. The purpose of this study was to examine multiple examples of cayley graphs through group theory, graph theory, and applications. In what follows we are concerned with using group theoretic methods to construct expander graphs, so we dont really care about all the extra structure that comes with a cayley graph. We will thus take our sets s hto satisfy the following properties. Although many specific cayley graphs have been shown to be hamiltonian, there are few general results. Pdf lovasz had posed a question stating whether every connected, vertex transitive graph has a hamilton path in 1969. While this is a lot, it doesnt seem unreasonably huge.
On nonhamiltonian circulant digraphs of outdegree three. If n number of vertices then the total number of unique hamiltonian circuits for a complete graph is 1. Cayley graphs of order 30p are hamiltonian article pdf available in discrete mathematics 31224 february 2011 with 76 reads how we measure reads. Cayley graphs are graphs associated to a group and a set of generators for that group there is also an associated directed graph. Hamiltonian circuits in graphs and digraphs springerlink.
Cayley graph, cayley digraph, hamilton circuit, hamilton cycle, generalized dihedral group. Letzter, hamiltonian circuits in cartesian products with a metaeyclic factor, ann. Hamilton connected cayley graphs on hamiltonian groups. Hamiltonian cycles and paths in cayley graphs and digraphs. Discrete mathematics 46 1983 4954 49 northholland hamiltonian circupi in cayley graphs dragan marui voike muc 12, 66000 koper, yugoslavia received 27 july i982 the following result is roved. A hamiltonian circuit is a circuit that visits every vertex exactly once. Check if you have access through your login credentials or your institution to get full access on this article. It is fairly easy to see that every connected cayley graph of an abelian group of order at least 3 has a hamiltonian circuit corollary 3. Cayley graph, hamiltonian cycle, normal connection set. How many circuits would a complete graph with 8 vertices have. A cayley graph is the subclass of vertextransitive graph, and.
Pdf hamiltonian cycles and paths in cayley graphs and. Some books call these hamiltonian paths and hamiltonian circuits. Hamiltonian paths of graphs, such as the graph above on the right, and to use that algorithm to draw conclusions about hamiltonian paths in the cayley digraphs of algebraic groups. One hamiltonian circuit is shown on the graph below. Combinatorics, graph theory and computing, congress, numer.
Unfortunately, the question of which graphs are hamiltonian does not seem to become signi cantly easier as a result of limiting the scope to closed graphs. Discrete mathematics elsevier discrete mathematics 156 1996 118 perspectives hamiltonian cycles and paths in cayley graphs and digraphs a survey stephen j. Ore o 1960 note on hamilton circuits, vol 67, no 1. Theorem 3 let g be a finite abelian group, and let s be any nonempty generating set for g. Received 24 august 1983 revised 24 june 1986 in this paper we will show that a hamiltonian cycle exists in the undirected cayley graph of. In the mathematical field of graph theory, a hamiltonian path or traceable path is a path in an undirected or directed graph that visits each vertex exactly once.
Roichman, random cayley graphs and expanders, random struc. Mendeley users who have this article in their library. Hamiltonian circuits, embeddings, faulttolerance, kary ncubes, npcompleteness. In this paper, we study the existence of hamiltonian paths and circuits in cays. Since the smallest nonabelian simple group has order ja5j 60, one can show that theorem 2 implies theorem 1 see section 3.
Find all hamilton circuits that start and end from a. Hamiltonian and eulerian graphs eulerian graphs if g has a trail v 1, v 2, v k so that each edge of g is represented exactly once in the trail, then we call the resulting trail an eulerian trail. Hamiltonian circuits mathematics for the liberal arts. Cayley graphs of finite abelian groups are guaranteed a hamiltonian. Determining whether such paths and cycles exist in graphs is the hamiltonian path problem, which is npcomplete. Faulttolerant hamiltonian laceability of cayley graphs generated by transposition trees. Cayley graphs and digraphs are introduced, and their importance and utility in group theory is formally shown. Since the smallest nonabelian simple group has order ja5j 60, one can show. Gallian, hamiltonian cycles and paths in cayley graphs and digraphsa survey, discrete math. An euler circuit is a circuit that reaches each edge of a graph exactly once.
Hamiltonian cycles in some easy cayley graphs university of. Nashwilliams let g be a finite graph with re 3 vertices and no loops or multiple edges. In particular, we give sufficient conditions for the group g to have the property that for every set s generating g, there is a hamiltonian circuit in cays. A comparison of methods for finding hamiltonian circuits.
Are there any edges that must always be used in the hamilton circuit. A graph is said to be eulerian if it contains an eulerian circuit. Hamiltonian and eulerian graphs university of south carolina. The problem is to find a tour through the town that crosses each bridge exactly once.
One such family of graphs is constructed using groups. Hamilton circuit is a circuit that begins at some vertex and goes through every vertex exactly once to return to the starting vertex. On hamiltonian cycles in cayley graphs of wreath products richard stong department of mathematics, harvard university, cambridge, ma 028, u. A hamiltonian cycle or hamiltonian circuit is a hamiltonian path that is a cycle. This circuit could be notated by the sequence of vertices visited, starting and ending at the same vertex.
Both of the t yp es paths eulerian and hamiltonian ha v e man y applications in a n um b er of di eren t elds. Hamiltonian cycles and paths in cayley graphs and digraphs a survey article pdf available in discrete mathematics 156. If the trail is really a circuit, then we say it is an eulerian circuit. On cayley digraphs that do not have hamiltonian paths. On hamiltonian circuits in cayley diagrams sciencedirect. Malkevitch, 8 this theory is named after leonhard euler, an outstanding mathematician during the 18th century. This paper compares three classes of algorithms for finding hamiltonian circuits in graphs. The traveling salesman problem is the problem of finding a hamiltonian circuit in a complete weighted graph for which the sum of the weights is a minimum. Most of the information found here is taken from john meiers, groups graphs and trees, 1. Half of the circuits are duplicates of other circuits but in reverse order, leaving 2520 unique routes. Notice that the circuit only has to visit every vertex once. Euler and hamiltonian paths and circuits lumen learning. Construction of hamiltonian circuits in cayley graphs.
However, none of these four graphs is a cayley graph. Hamiltonian paths in cartesian powers of directed cycles dave witte department of mathematics oklahoma state university stillwater, ok 74078. Starling, hamiltonian cycles in cayley color graphs of semidirect products, proc. This inspired many people to conjecture that every connected cayley graph with more than two vertices has a hamiltonian ciccuit. Specifically, the following result lists up to isomorphism the only known examples of connected, nonhamiltonian cayley digraphs, such that and. If there exists suc h w e ould also lik an algorithm to nd it. Two of the classes are exhaustive search procedures and this study finds them to have an exponential dependence on the size of the graph. Pdf hamiltonconnected cayley graphs on hamiltonian groups. We gave background material on groups and graphs and gave numerous examples of cayley graphs and digraphs.
There are several other hamiltonian circuits possible on this graph. Hamiltonian cycles in cayley graphs, discrete math. Hamilton path is a path that contains each vertex of a graph exactly once. Z m x z n has a hamiltonian circuit when n divides m. Every finite cayley graph with at least three vertices contains a hamilton cycle. Hamiltonian circuit when m and n are relatively prime and greater than 1. The construction of hamiltonian circuits represented by sequences of the weights of edges is. Eac h of them asks for a sp ecial kind of path in a graph.
However, the closure procedure has a somewhat cumulative e ect on many graphs. Faulttolerant embeddings of hamiltonian circuits in k. Lovasz conjecture claims that every connected cayley graph contains a hamiltonian path. Cayley graphs with commutator subgroup of order 2p are hamiltonian 3 we use g0to denote the commutator subgroup g. Pdf a survey on hamiltonicity in cayley graphs and digraphs on. The edge representation of circuits in ndimensional binary hypercubes, which is put forward in 1, is extended to a number of other cayley graphs that differ in the finite groups determining the sets of their vertices and in the permutations of the numbers of vertices, which prescribe their edges. Construct a graph that has both an euler and a hamiltonian circuit. Hamiltonicity of cayley graphs open problem garden. American mathematical monthly, mathematical association of america, p 55. The regions were connected with seven bridges as shown in figure 1a. A complete graph with 8 vertices would have 5040 possible hamiltonian circuits. Another goal of the project was to write a program in maple that would draw a cayley digraph in a manner that is useful and educational to a viewer. To state this result, we introduce a bit of notation. These graphs are called cayley graphs and they are the subject of this chapter.
Hamiltonian circuits in cayley graphs discrete mathematics. Hamilton circles in cayley graphs the electronic journal of. Two vertices are adjacent if they are joined by an edge. Gallianb a department of mathematics, university of pittsburgh, johnstown, johnstown, pa 15904, usa b department of mathematics, university of minnesotaduluth, duluth, mn 55812, usa received 16 august 1994. Apparently, only four nontrivial connected vertex symmetric aphs without hamiltonian circuits are known i. In fact, it was known that only five vertextransitive graphs exist without a hamiltonian cycle which do not belong to cayley graphs. Hamiltonian circuits in cayley graphs sciencedirect. Cayley graphs abstract there are frequent occasions for which graphs with a lot of symmetry are required. Chapter 10 eulerian and hamiltonian p aths circuits this c hapter presen ts t w o ellkno wn problems.
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