Point-symmetric graphs with a prime number of points

Turner, James

doi:10.1016/s0021-9800(67)80003-6

Cited by 132 publications

(55 citation statements)

References 3 publications

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“…Contrary to the claim of Turner [13], this graph is, in fact, a circulant with symbol (6,{2,3,41). We now determine those vertex-transitive graphs which are circulants.…”

Section: Circulantsmentioning

confidence: 56%

“…A topic of recent interest in the literature involves the characterization of vertex-transitive graphs [2,3,5,6,[8][9][10][11]13]. In section 2, we present two such characterizations.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, the problem of characterizing vertextransitive graphs in a more graph-theoretical manner appears to be very difficult. In [13], Turner provided a partial solution to the problem by showing that a graph with a prime number of nodes is vertex-transitive if and only if it is a circulant (or, equivalently, a starred polygon).…”

Section: Introductionmentioning

confidence: 99%

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Circulants and the Characterization of Vertex-Transitive Graphs

Leighton

1983

J. RES. NATL. BUR. STAN.

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In this paper, we extend the notion of a circulant to a broader class of vertex.transitive graphs, which we call multidimensional circulants. This new class of graphs is shown to consist precisely of those vertextransitive graphs with an automorphism group containing a regular abelian subgroup. The result is proved using a theorem of Sabidussi which shows how to recover any vertex-transitive graph from any transitive subgroup of its automorphism group. The approach also allows a short proof of Turner's theorem that every vertex-transitive graph on a prime number of nodes is a circulant.

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“…Contrary to the claim of Turner [13], this graph is, in fact, a circulant with symbol (6,{2,3,41). We now determine those vertex-transitive graphs which are circulants.…”

Section: Circulantsmentioning

confidence: 56%

“…A topic of recent interest in the literature involves the characterization of vertex-transitive graphs [2,3,5,6,[8][9][10][11]13]. In section 2, we present two such characterizations.…”

Section: Introductionmentioning

confidence: 99%

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Circulants and the Characterization of Vertex-Transitive Graphs

Leighton

1983

J. RES. NATL. BUR. STAN.

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“…A large number of articles directly or indirectly related to this problem (for the list of relevant references and a detailed description of the status of this problem see [67]), have appeared in the literature, affirming the existence of such paths in some special vertex-transitive graph and, in some cases, also the existence of Hamilton cycles. It is known that connected vertex-transitive graphs of order kp, where k ≤ 4, and p j , where j ≤ 4, and 2p 2 , where p is a prime, (except for the Petersen graph and the Coxeter graph) contain a Hamilton cycle, whereas for connected vertex-transitive graphs of order 5p and 6p it is known that they contain Hamilton paths (see [1,2,3,4,5,6,8,9,10,11,26,65,68,81,82,83,89,90,119]). Particular attention has been given to Cayley graphs.…”

Section: Hamilton Paths and Cyclesmentioning

confidence: 99%

Recent trends and future directions in vertex-transitive graphs

Kutnar

Marušič

2008

Ars Math. Contemp.

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A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade.Keywords: Vertex-transitive graph, arc-transitive graph, half-arc-transitive graph, Hamilton cycle, Hamilton path, semiregular group, (im)primitive group.Math. Subj. Class.: 05C25, 20B25 1 In the beginning Vertex-transitive graphs, that is, graphs whose automorphisms groups acts transitively on the corresponding vertex sets, have been an active topic of research for a long time now. Much of this interest is due to their suitability to model scientific phenomena when symmetry is an issue. It is the aim of this article to discuss recent developments surrounding two well known open problems in vertex-transitive graphs -the Hamilton path/cycle problem and the semiregularity problem -as well as the general question addressing structural properties of arc-transitive and half-arc-transitive graphs. In doing so we will also try to contemplate possible directions this area of research is likely to take in the near future. * Supported by "Agencija za raziskovalno dejavnost Republike Slovenije", research program P1-0285.E-mail addresses: klavdija.kutnar@upr.si (Klavdija Kutnar), dragan.marusic@upr.si (Dragan Marušič) Copyright c 2008 DMFA -založništvo K. Kutnar, D. Marušič: Recent Trends and Future Directions in Vertex-Transitive Graphs 113The article is organized as follows. In Section 2 we discuss the problem, posed by the second author in 1981 (see [79]) who asked if it is true that a vertex-transitive digraph contains a nonidentity automorphism with all orbits of equal length, in short, a semiregular automorphism. Section 3 gives a quick overview of the problem, posed by Lovàsz in 1969 (see [72]), who asked if it is true that every connected vertex-transitive graph contains a Hamilton path, thus motivating a great deal of research into vertex-transitive graphs in the following decades. In Section 4, imprimitivity, one of the most fundamental concepts in the theory of permutation groups, is considered with special emphasis given to a recently developed theory which links the existence of blocks of imprimitivity in vertex-transitive graphs, having an abelian semiregular subgroup of automorphisms, to certain conditions that need to be satisfied in the corresponding quotient graph relative to the orbits of such a subgroup. Finally, in Section 5 we deal with some recent structural results on arc-transitive and half-arc-transitive graphs, as well as their link to some open problems in the theory of configurations.For group-theoretic terms not defined here we refer the reader to [124]. Semiregularit...

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“…It is known that such a graph is isomorphic to a circulant Cay(Z p , S), whose automorphism group is generated by the left translations and the automorphisms of Z p that preserve the connection set S (see Turner [122]). By the preceding result, these graphs are also cores, and thus have no endomorphisms other than their automorphisms.…”

Section: Cores Of Vertex-transitive Graphsmentioning

confidence: 99%

Graph homomorphisms: structure and symmetry

1997

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This paper is the first part of an introduction to the subject of graph homomorphism in the mixed form of a course and a survey. We give the basic definitions, examples and uses of graph homomorphisms and mention some results that consider the structure and some parameters of the graphs involved. We discuss vertex transitive graphs and Cayley graphs and their rather fundamental role in some aspects of graph homomorphisms. Graph colourings are then explored as homomorphisms, followed by a discussion of various graph products.

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Point-symmetric graphs with a prime number of points

Cited by 132 publications

References 3 publications

Circulants and the Characterization of Vertex-Transitive Graphs

Circulants and the Characterization of Vertex-Transitive Graphs

Recent trends and future directions in vertex-transitive graphs

Graph homomorphisms: structure and symmetry

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