Following a problem posed by lovasz in 1969, it is believed that every connected vertextransitive graph has a hamilton path. Snarks asnarkis a connected, bridgeless cubic graph with chromatic. Royle observed that most vertextransitive graphs of order at most are cayley graphs, and this led mckay and praeger to conjecture 1994 that most vertextransitive graphs are cayley graphs, see. The planar cubic cayley graphs mathematics tu graz. We next classify all 2extendable cubic nonbipartite cayley graphs and obtain that. Cayley graphs and digraphs are introduced, and their importance and utility in group theory is formally shown. The motivation for the testing of cayley graphs for hamiltonicity was the conjecture that states that every connected cayley graph is hamiltonian.
These graphs are called cayley graphs and they are the subject of this chapter. A cubic cayley graph based on an abelian group of girth 3 is either k4, or the 3 sided prism. Elena konstantinova lecture notes on some problems on. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Q on hamiltonicity of vertextransitive graphs and digraphs of order p4. As an application, a classification of connected cubic vertextransitive bicayley graphs over abelian groups is given, and using this, a problem posed in zhou and feng 2012 regarding the cayley property of a class of graphs is solved.
Among these, the cubic cayley graphs with up to 40320 vertices were tested for various properties including hamiltonicity and diameter. The wellknown conjecture that there are no snarks amongst cayley graphs is considered. Despite the above, i am here to suggest that polynomial algorithms may exist that will supply a correct answer but only \most of the time. This is shown here to be true for cubic cayley graphs. 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. Hamiltonicity problem is hard, the snark problem is hard too, but should be easier to deal with. Searching for an innovative approach to hamilton cycles. In this article we consider the hamiltonicity problem for cubic cayley graphs arising from groups having a 2,s,3presentation, that is, for groups g ha,ba2 1,bs 1,ab3 1,etc. Note that a cubic cayley graph must belong to one of two types. Graph theory and cayleys formula university of chicago. One such family of graphs is constructed using groups. In fact, we describe all 3connected planar cubic cayley graphs with the. A more detailed account on cubic hamiltonian graphs will be presented in a forthcoming paper 7.
A group action gon a space xis free if, for any xin x, and any gand g 0in g, gx g0ximplies g g. A hamiltonian cycle or hamiltonian circuit is a hamiltonian path that is a cycle. 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. The planar cubic cayley graphs agelos georgakopoulos. Hamilton cycles in graphs embedded into surfaces roman nedela university of west bohemia, czech rep. Semiregular automorphisms in vertextransitive graphs hamiltonicity of vertextransitive graphs most recent results for cubic cayley graphs glover, kutnar, malni c, dm, 200711. Cubic bicayley graphs over abelian groups sciencedirect. Vertextransitive graphsi a graph xv,eis vertextransitive if for any pair of vertices u,vthere exists. Rossomakhine detecting nonhamiltonicity in a cubic graph. Over 10 million scientific documents at your fingertips. The research is motivated indirectly by the long standing conjecture that all. This is shown here to be true for cubic cayley graphs arising from groups having a 2, s, 3presentation, that is, for groups g. This turns out to be a rich class, comprising several in. Durnberger,connected cayley graphs of semidirect products of cyclic groups of.
Note that a 3connected planar graph has an essentially unique embedding see theorem 3. Find materials for this course in the pages linked along the left. Independent sets and their complements in cubic graphs restriction to cyclically 4edgeconnected cubic graphs. Cayley graphs abstract there are frequent occasions for which graphs with a lot of symmetry are required. Elena konstantinova lecture notes on some problems on cayley. Hamiltonian cycles for this particular class of cubic cayley graphs. The hypercube or n dimensional cube hn is the graph with vertex set. When a cubic graph is hamiltonian, lcf notation allows it to be represented concisely. Hamiltonicity of cubic cayley graphs received september 1, 2005 and in revised form september 11, 2006 abstract. In this section, we generalize the idea of a single generator of a group to a whole set of generators of a group. Most of the information found here is taken from john meiers, groups graphs and trees, 1. Speci cally, we are going to develop cayley graphs and schreier diagrams, use them to study various kinds of groups, and from there prove some very deep and surprising theorems from abstract algebra. Hamiltonicity of cubic cayley graphs henryglover draganmarusic1 departmentofmathematics imfm,universityofljubljana ohiostateuniversity anduniversityofprimorska,koper u.
Skoviera although hundreds of papers deals with the problem of existence of a hamiltonian cycle in a graph, there is a luck of results on the hamiltonicity of cubic graphs. The aim of the current paper is a classification of the planar cubic cayley graphs of connectivity 2, yielding an explicit presentation and embedding for each of those graphs. Although many specific cayley graphs have been shown to be hamiltonian, there are few general results. In this paper, some basic properties and the automorphisms of bicayley graphs are explored. Pdf following a problem posed by lov\asz in 1969, it is believed that every connected vertextransitive graph has a hamilton path. A cayley graph is the subclass of vertextransitive graph, and in view of. Remember, a cyclic group has a single generator and is isomorphic to either z if it is of in. These results are available on the internet in easy to read tables. Cayley vertextransitive graph is 2extendable when it is of minimum degree at least.
If a cubic graph is chosen uniformly at random among all nvertex cubic graphs, then it is very likely to be hamiltonian. A survey on hamiltonicity in cayley graphs and digraphs on. Pdf a survey on hamiltonicity in cayley graphs and digraphs on. Robinson abstract this thesis investigates the generation of nonisomorphic simple cubic cayley graphs. 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. Class 1 cubic cayley graphs are 2generated cayley graphs for which just one generator is an involution. Unravelling the miniature rubiks cube through its cayley graph daniel bump1 and daniel auerbach2 october, 2006 1 introduction there are many interesting questions one can ask about a highly symmetric graph. Conditions on hamiltonicity, discrete mathematics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Hamiltonicity of cubic cayley graphs european mathematical. Thus each automorphism of x is a permutation of the vertex set. In what follows we list statements establishing hamiltonicity of.
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. Enumeration, isomorphism and hamiltonicity of cayley. The ones marked may be different from the article in the profile. The planar cubic cayley graphs agelos georgakopoulos u of ottawa tu graz dartmouth college, 12. Here we are only interested in cubic cayley graphs. Semiregular subgroups of transitive permutation groups. Combined with 9 this yields a complete description of all planar cubic cayley graphs. On hamiltonicity of vertextransitive graphs and digraphs of order p 4. Since the smallest nonabelian simple group has order ja5j 60, one can show that theorem 2 implies theorem 1 see section 3. Cayley graphs week 5 mathcamp 2014 today and tomorrows classes are focused nthe interplay of graph theory and algebra. 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. A vertextransitive graph of order n is a cayley graphprovided it has a transitive subgroup of automorphisms of order n.
Pdf hamilton cycles in cubic cayley graphs on dihedral groups. Generation of nonisomorphic cubic cayley graphs by bijaya rath under the direction of robert w. Hamiltonconnected cayley graphs on hamiltonian groups. Shields, ian beaumont hamilton cycle heuristics in hard graphs under the direction of professor carla d.
Cubic cayley graphs and snarks vanderbilt university. Cubic cayley graphs and snarks klavdija kutnar university of primorska nashville, 2012. Cayley graph with jxj kand is called a kgenerated cayley graph. Pdf lovasz had posed a question stating whether every connected. Following a problem posed by lovasz in 1969, it is believed that every finite connected vertextransitive graph has a hamilton path. The main advantage in using cayley graphs as models for interconnection net. Determining whether such paths and cycles exist in graphs is the hamiltonian path problem, which is npcomplete. Given a connected graph x, a subset f ex is called cycleseparatingif x f is disconnected and at least two of its components contain cycles.
Combining the theory of cayley maps with the existence of certain kinds of independent sets of vertices in arctransitive graphs, some new partial results are obtained suggesting promising future research directions in regards to this conjecture. Keywords hamiltonian cycle cayley graph cubic arctransitive graph. The connectivity of a graph g is the smallest cardinality of a set of vertices separating g. We classify the planar cubic cayley graphs of connectivity 2, providing an explicit presentation and embedding for each of them. 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. We say that x iscyclically kedgeconnectedif no set of fewer than k edges is cycle. For cubic cayley graphs on dihedral groups, the conjecture was proved to be true. Read connected graphs as subgraphs of cayley graphs. This cited by count includes citations to the following articles in scholar. The vertices in such graphs correspond to processing elements, memory modules, and the edges correspond to communication lines. Savage in this thesis, we use computer methods to investigate hamilton cycles and paths in several families of graphs where general results are incomplete, including kneser graphs, cubic cayley graphs and the middle two levels graph.
325 1638 759 850 1093 1286 1420 1144 991 1190 1271 1042 1450 944 615 1634 706 1048 1178 960 322 124 921 327 226 752 263 144 1397 531 348 1161 1264 1165 145