Graph theory and cayleys formula university of chicago. In this paper we prove that the analogue of the chenquimpo theorem holds for cayley graphs on hamiltonian groups. We shall ignore labels and orientation of edges and treat. Denote by rg and mg the number of abelian and nonabelian composition. Pdf lovasz had posed a question stating whether every connected, vertex transitive graph has a hamilton path in 1969. Let g be a hamiltonian group and s a generating set of g.
Finally, we show that all cayley graphs of abelian. An important example of a group action is the action of gon the set of elements in g, denoted gy fg2gg, by group multiplication on the left. Structural theorem on plane graphs with application to the entire coloring number. 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. The cayley graph c g, s has the elements of g as nodes, and two nodes x and y are adjacent if and only if y x. One such family of graphs is constructed using groups. Enumeration, isomorphism and hamiltonicity of cayley. Hamiltonicity of cayley graphs open problem garden. Hamiltonicity of cubic cayley graphs article pdf available in journal of the european mathematical society 94 october 2005 with 52 reads how we measure reads. A group action gon a space xis free if, for any xin x, and any gand g 0in g, gx. As mentioned in the introduction, cayley graphs are important examples of vertex transitive graphs. That is, as the order n of a groupg approaches infinity, the ratio of the number of hamiltonian cayley graphs ofg to the total number of cayley graphs ofg approaches 1.
The motivation for the testing of cayley graphs for hamiltonicity was the conjecture that states that every connected cayley graph is hamiltonian. Finally, we show that all cayley graphs of abelian groups have. It is then proven that every cayley di graph of a dedekind group has a hamiltonian path. It is proven that every connected cayley graph x, of valency at least three, on a hamiltonian group is either hamilton laceable when x is bipartite, or hamilton connected when x is not bipartite. Tuttes theorem every cubic graph contains either no hc, or at least three examples of hamiltonian cycles in cayley graphs of s n. Elena konstantinova lecture notes on some problems on cayley. These results are available on the internet in easy to read tables. Searching for an innovative approach to hamilton cycles. Almost all cayley graphs are hamiltonian springerlink. Hamilton cycles in 2,s,3cayley graphs hamilton cycles in graph covers hamiltonicity of vtg of order a product of two primes searching for an innovative approach to hamilton cyclespaths problem in. A hamiltonian cycle, hamiltonian circuit, vertex tour or.
Most of the information found here is taken from john meiers, groups graphs and. Cayley graphs abstract there are frequent occasions for which graphs with a lot of symmetry are required. This is shown here to be true for cubic cayley graphs arising. Assuming the conjecture holds true, to prove the hamiltonicity of cubic cayley graphs it is sufficient to examine cubic cayley graphs of girth at most 6. In mathematics, a cayley graph, also known as a cayley colour graph, cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. In fact, a much stronger result about the hamiltonicity of cayley graphs on abelian groups is known.
Hamiltonconnected cayley graphs on hamiltonian groups core. Most of the information found here is taken from john meiers, groups graphs and trees, 1. Cayley graph, automorphism, transposition, cycle, conjugate, hamiltonian cycle. Speci cally, we are going to develop cayley graphs and schreier. The main advantage in using cayley graphs as models for interconnection net. That is, as the order n of a groupg approaches infinity, the ratio of the number of hamiltonian cayley. It is then proven that every cayley digraph of a dedekind group has a hamiltonian path. Cayley graphs and digraphs are introduced, and their importance and utility in group theory is formally shown. Then jsj b3n 22cand more precisely, the following hold.
Max dehn in his unpublished lectures on group theory from 190910 reintroduced cayley graphs under the name gruppenbild group diagram, which led to the geometric group theory of today. Unravelling the miniature rubiks cube through its cayley graph. Tripi a masters thesis submitted to the graduate college of missouri state university in partial ful llment of the requirements for the. Using this result we prove further that every connected non.
Cayley colour diagrams were generalized to schreier coset diagrams by o. A group action gon a space xis free if, for any xin x, and any gand g 0in g, gx g0ximplies g g. 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. Snarks asnarkis a connected, bridgeless cubic graph with chromatic. A vertextransitive graph is acayley graphprovided there exists a. A graph that contains a hamiltonian path is called a traceable graph.
Cayley graphs of groups and their applications by anna m. Pdf a survey on hamiltonicity in cayley graphs and. Cayley tetravalent metacirculant graph is isomorphic to a union of disjoint copies of a graph in one of the families. This is shown here to be true for cubic cayley graphs arising from groups having a 2, s, 3presentation, that is, for groups g. The decision to focus on the twogenerator group generated just by rotations of two adjacent faces is motivated not only by the desire to obtain a cayley graph of. Witte, hamilton cycles in cayley graphs with cyclic commutator subgroup, annals of discrete mathematics, vol. We show that if is any generating set of, then there is a hamiltonian cycle in the corresponding cayley graph cay 1. Let g be a finite group with identity element 1, and let s.
Cayley graphs admits a hamilton circle, see theorem 4. Following a problem posed by lovasz in 1969, it is believed that every connected vertextransitive graph has a hamilton path. A cayley colour diagram is a directed graph with coloured edges cf. Tripi a masters thesis submitted to the graduate college of missouri state university in partial ful llment of the requirements for the degree of master of science, mathematics july 2017 approved. Suppose that is a finite group, such that, where is prime. From this point on, we consider only cayley graphs. Grinbergs formula lovasz and babai conjectures for vertextransitive graphs diracs theorem. Cayley graphs were first considered for finite groups by arthur cayley in 1878.
A vertextransitive graph of order n is a cayley graphprovided it has a transitive subgroup of automorphisms of order n. In particular, it is unknown whether every undirected cayley. In spite of the difficulty in proving this conjecture, we show that almost all cayley graphs are hamiltonian. In this short note we give an introduction to some elementary properties of cayley graphs. Assuming the conjecture holds true, to prove the hamiltonicity of cubic cayley graphs it is. Pdf a survey on hamiltonicity in cayley graphs and digraphs on.
Hamiltonconnected cayley graphs on hamiltonian groups. Among these, the cubic cayley graphs with up to 40320 vertices were tested for various properties including hamiltonicity and diameter. Schreier in 1927, and both were investigated as graphs in a20. Morris, 2generated cayley digraphs on nilpotent groups have hamiltonian paths, preprint. Semiregular subgroups of transitive permutation groups. Hamiltonicity, basic results tuttes counter example length of the longest path in a planar graph. Cayley graphs week 5 mathcamp 2014 today and tomorrows classes are focused nthe interplay of graph theory and algebra. The decision to focus on the twogenerator group generated just by rotations of two adjacent faces is motivated not only by the desire to obtain a cayley graph of convenient size, but also by a circumstance described by singmaster 6 the twogenerator group is a particularly interesting group. A survey on hamiltonicity in cayley graphs and digraphs on.
Hamiltonian cycles of cayley graphs have been extensively studied, although general results are relatively few. Cayley graphs and coset diagrams 1 introduction let g be a. Although many specific cayley graphs have been shown to be hamiltonian, there are few general results. Following a problem posed by lovasz in 1969, it is believed that every finite connected vertextransitive graph has a hamilton path. Pdf a survey on hamiltonicity in cayley graphs and digraphs. Cayley tetravalent metacirculant graph is isomorphic to a union of disjoint copies of a graph in. Enumeration, isomorphism and hamiltonicity of cayley graphs. Cayley graphs formed by conjugate generating sets of s. These graphs are called cayley graphs and they are the subject of this chapter. Supereulerian graphs, hamiltonicity of graphs and several extremal problems in graphs pour une soutenance en 27 septembre 20 composition du jury. These cayley graphs range in size up to 5040, and include a number for which hamiltonicity or nonhamiltonicity has not been determined. Hamiltonicity of vertextransitive graphs semiregular automorphisms in vertextransitive graphs cayley graphs cayley graph given a group g and a subset s of g nf1g, s s 1, thecayley graphcayg. This survey will describe the current status of the search for hamiltonian cycles and paths in cayley graphs and digraphs on different groups, and discuss the future. One exception is the theorem by pak and radoicic pr.
The vertices in such graphs correspond to processing elements, memory modules, and the edges correspond to communication lines. Encoding 5 5 a forest of trees 7 1 introduction in this paper, i will outline the basics of graph theory in an attempt to explore cayleys formula. Elena konstantinova lecture notes on some problems on. Lecture notes on some problems on cayley graphs elena konstantinova koper, 2012. Chen and quimpo 4 investigated cayley graphs on hamiltonian groups and proved the following result. Similarly, the connectivity of a cayley graph on a nite abelian group is easy to determine, while. Cubic cayley graphs and snarks klavdija kutnar university of primorska nashville, 2012.
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