Every symmetric graph without isolated vertices is vertex transitive, and every vertex transitive graph is regular. As a result, there are 11 sporadic and one infinite family of such graphs, of which the. The petersen graph on 10 vertices is the smallest example of a vertextransitive graph which is not a cayley graph. For example, the skeleton of the c 60 molecule is a. On the decomposition of vertextransitive graphs into multicycles. Every symmetric graph without isolated vertices is vertextransitive, and every vertextransitive graph is regular. Abstract the petersen graph on 10 vertices is the smallest example of a vertex transitive graph which is not a cayley graph. Identifying codes in vertex transitive graphs sylvain gravier, aline parreau, sara rottey, leo storme and elise vandomme icgt 2014 grenoble. However, not all vertextransitive graphs are symmetric for example. Vertextransitive graph wikimili, the best wikipedia reader.
Pdf on isomorphisms of vertextransitive graphs researchgate. Compression of vertex transitive graphs ucf cs university of. A graph is vertextransitive if and only if its graph complement is, since the group actions are identical. A 27vertex graph that is vertextransitive and edge. A study of necessary and sufficient conditions for vertex transitive. It is easy to test whether a given vertextransitive graph is a cayley graph for some group. September 22, 1982 in this paper, we prove that every vertex transitive graph can be expressed as the edgedisjoint union of symmetric graphs. Recently, godsil 2 has determined those kneser graphs kn, k which.
In the mathematical field of graph theory, a vertextransitive graph is a graph g in which, given. Narsingh deo, y and aurel cami y abstract we consider the lossless compression of vertex transitive graphs. He showed that we can partition a vertextransitive graph into isomorphic subgraphs, called atomic parts, such that the vertextransitivity is. Vertextransitive graphs which are not cayley graphs ii article pdf available in journal of the australian mathematical society 5601 february 1994 with 247 reads how we measure reads.
Pdf cubic vertextransitive graphs of order 2 jinxin. Leader, imre 2001, a conjecture concerning a limit of noncayley graphs pdf, journal of algebraic combinatorics, 14 1. Thus while all vertices and edges of this graph are similar, there are no edgereversing automorphisms. Introduction to vertextransitive graphs of primepower order. Vertextransitive graphs that fail this test are relatively rare, the petersen graph being a famous example. The pseudorhombicuboctahedron which is not isogonal demonstrates that simply asserting that all vertices look the same is not as restrictive as the definition used here, which involves the group of isometries preserving the polyhedron. Let g be a permutation group acting on the nonempty set v. The graph g is vertex transitive if its group of automorphisms acts transitively on v see 7 for a careful account of such symmetry conditions. Circulants and the characterization of vertextransitive graphs. Compression of vertex transitive graphs bruce litow. Vertextransitive is a synonym borrowed from modern ideas such as symmetry groups and graph theory. Watkins 71 showed that nggg 2 which means that the vertex connectivity of any given vertex transitive graph will not be too small. A complete classi cation is given for nite vertex primitive and vertex biprimitive s transitive graphs for s 4.
Pdf cores of vertex transitive graphs semantic scholar. Hahn and tardif have shown that for vertex transitive graphs, the size of the core must divide the size of the graph. Leighton massachusetts institute of technology, cambridge, ma 029 accepted. A cis graph is a graph in which every maximal stable set and every maximal clique intersect. More confusingly still, arc transitive graphs and therefore in fact transitive graphs for are sometimes called symmetric graphs godsil and royle 2001, p. If bis a bipartite vertextransitive graph such as the 4cycle, then a. A 1 transitive graph is simply called an arc transitive graph or even a transitive graph. E is called vertex transitive if for every pair of vertices x. A core of a graph x is a vertex minimal subgraph to which x admits a homomorphism. We shall suppose graphs are vertex transitive except where otherwise stated.
Not every vertex transitive graph is a cayley graph. Finite vertex transitive graphs include the symmetric graphs such as the petersen graph, the heawood graph and the vertices and edges of the platonic solids. Sabidussi 5 has characterized vertextransitive graphs in a way which shows their close relation to cayley graphs. A note on coloring vertextransitive graphs daniel w. In this paper we show that all locally finite, 2 transitive graphs and all 1 transitive graphs of prime degree are accessible. However, it is usually not easy to compute whether given vertextransitive graphs are cayley graphs. Sabidussi 5 has characterized vertex transitive graphs in a way which shows their close relation to cayley graphs. Selfcomplementary vertextransitive graphs bulletin of the. Hitting times for random walks on vertextransitive graphs. A graph is said to be vertex transitive noncayley if its full automorphism group acts transitively on its vertices and contains no subgroups acting regularly on its vertices. Oct 29, 20 i describe the 5 vertex transitive nonhamiltonian graphs which are known.
Adjacencylist digraph representation maintain vertexindexed array of lists. The finite vertex primitive and vertex biprimitive s transitive graphs for s 4 cai heng li abstract. Doyle version dated 1985 gnu fdly abstract i describe a 27 vertex graph that is vertex transitive and edge transitive but not 1 transitive. We prove that every locally finite, vertex transitive graph with at least one thick end has a thick end with a 2way infinite geodesic, while no thin end has a 2way infinite geodesic. Adjacency matrix digraph representation maintain a twodimensional v v boolean array. A useful general strategy for the construction of interesting families of vertextransitive graphs is to begin with some family of transitive permutation groups and to. A graph undirected, without loops or multiple edges is said to be vertex transitive if its automorphism group acts transitively on the set of vertices. The graph x is halfarc tr ansitive if it is vertextransitive and edgetransitive, but not arctransitive. Every vertex transitive graph of order p3 is a cayley graph. Then all vertices in g are cutvertices because g is vertex transitive.
We conjecture that every vertextransitive graph satis es. Pdf the isomorphism problem of cayley graphs has been well studied in the literature, such as characterizations of ci dcigraphs and ci. A major class of vertex transitive graphs is formed by cayley graphs, whic h repre sent groups in a very natural w ay. The finite cayley graphs such as cubeconnected cycles are also vertex transitive, as are the vertices and edges of the archimedean solids though only two of these are symmetric. The petersen graph on 10 vertices is the smallest example of a vertex transitive graph which is not a cayley graph. We consider the lossless compression of vertex transitive graphs.
Search on vertextransitive graphs by lackadaisical quantum walk. In this paper, we observe that these are all vertex transitive graphs, and when there is a unique marked vertex, the optimal weight of the selfloop equals the degree of the loopless graph divided by the total number of vertices. A graph is wellcovered if all its maximal stable sets are of the same size, cowellcovered if its complement is wellcovered, and vertex transitive if, for every pair of vertices, there exists an automorphism of the graph mapping one to the other. A graph isvertextransitiveif its automorphism group aut is transitive on v, the vertex set of.
For generalized petersen graphs its easy to construct required automorphisms there are obvious rotational symmetries taking any outer vertex to any other outer vertex, and any inner vertex to any other inner vertex. Pdf vertextransitive graphs which are not cayley graphs ii. Cubic vertextransitive noncayley graphs of order 12 p. Based on this result we prove that a vertex transitive graph is cis if and only if it admits a strong clique and a strong independent set. A 27 vertex graph that is vertex transitive and edge transitive but not l transitive peter g. The classi cation involves the construction of new 4 transitive graphs, namely a graph of valency 14. In other words, a graph is vertex transitive if its automorphism group acts transitively upon its vertices. For the selfcomplementary vertextransitive graphs of order a product of two primes, we give a complete classi.
Cranston landon raberny april 25, 2014 abstract we prove bounds on the chromatic number. Every vertex transitive graph of order p2 is a cayley graph. The aim of the present paper is to illustrate concepts and methods employed while working in the area of vertextransitive graphs or vtgraphs for short, through recently obtained significant results. I describe the 5 vertextransitive nonhamiltonian graphs which are known. Pdf cores of vertex transitive graphs david roberson. Intuitively, a graph is vertex transitive if there is no structural i. We propose that this holds for all vertex transitive graphs with a unique marked vertex. All of these theorems reaffirm the idea of using vertex transitive graphs when designing networks. This graph is vertex transitive but has no automorphism of order eight and thus cannot be a circulant. In this, the first of a series of papers, we present a. Every undirected graph is a digraph happens to have edges in both directions bfs is a digraph algorithm visits vertices in increasing distance from s put s onto a fifo queue. Vertex transitive graphs and their arctypes marston conder, toma z pisanski, and arjana zitnik 6 may 2015 abstract let x be a nite vertex transitive graph of valency d, and let a be the full automorphism group of x. Directed graphs princeton university computer science.
Pdf strong cliques in vertextransitive graphs semantic. I also pose two problems for the interested viewer to solve. Pdf vertextransitive graphs which are not cayley graphs, i. Vertextransitive graphs and accessibility sciencedirect. An undirected graph g v,e is called vertex transitive if for every pair of vertices x. Search on vertextransitive graphs by lackadaisical.
University of primorska, slovenia joint work with edward dobson, martin milani. A 27vertex graph that is vertextransitive and edgetransitive but not. Note that the valency of a halfarctransitive graph is necessarily even. Let x be a finite vertextransitive graph of valency d, and let a be the full automorphism group of x. However, it is usually not easy to compute whether given vertex transitive graphs are cayley graphs. We consider the problem of determining the orders of such graphs. As for the vertex connectivity of any vertex transitive graph g, m.
For the selfcomplementary vertex transitive graphs of order a product of two primes, we give a complete classi cation of these graphs. We consider the problem of what orders such graphs have. In this paper, a complete classification of cubic vertex transitive noncayley graphs of order 12p, where p is a prime, is given. On the decomposition of vertex transitive graphs into multicycles f.
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