Given a graph g, an independent set ig is a subset of the vertices of g such that no two vertices in ig are adjacent. The edge metric dimension of the generalized petersen. The number of elements in the set of connected cubic graphs on vertices grows rapidly with. A graph that has fascinated graph theorists over the years because of its appearance as a counterexample in so many areas of the subject. The petersen graph australian mathematical society lecture. The petersen graph is the complement of the line graph of. A borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs pdf. We can see that this is the case by using vertextransitivity hence only one vertex needs to be considered and observing the following drawing of the petersen graph. Watkins skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Arora, novi high school the strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor. The groups of the generalized petersen graphs volume 70 issue 2 roberto frucht, jack e.
In order to state the main result we need to introduce the graph and two 2parametric families of cubic, connected graphs. If uv 62eg, then there are m vertices x so that ux 2eg and xv 2eg. In this paper, we prove that the generalized petersen graphs p n, 1,p n, na. However, removing any vertex leaves a hamiltonian graph, so the petersen graph is described as hypohamiltonian.
On the domination number of the generalized petersen graphs. Thus, it can be described as the graph with as vertex set the pairs from a 5set, where two pairs are joined when they are disjoint. May, 20 kuratowskis theorem tells us that if k5 is a subdivision of a graph then that graph is not planar. The petersen graph g is the simple graph with 10verties and 15edges. On minimum vertex cover of generalized petersen graphs. The petersen graph does not have a hamiltonian cycle. Many additional facts about the petersen graph can be found in 4, 8. We determine when the generalized petersen graphs g p n, k are superconnected and super edgeconnected, and show that their superconnectivity and their superedgeconnectivity are both. Automorphisms of graphs math 381 spring 2011 an automorphism of a graph is an isomorphism with itself.
On the partition dimension of caveman graph, gneralized. Following up on petersen graph with new tikz graph library, i tried to label the vertices with something different than numbers. The main purpose of this paper is to prove two results about the spectrum of the generalised petersen graphs. In this case, the dual graph is cubic and bridgeless, so by petersens theorem it has a matching, which corresponds in. Vg, such that u v is an edge if and only if uvis an edge. A solution of the hamiltons game is illustrated by the heavy edges in figure 1.
We present two algorithms which between them lead to the determination of upper and lower bounds on the domination numbers of these graphs, and. On the metric dimension of generalized petersen graphs article pdf available in ars combinatoria waterloo then winnipeg 105. We say preserves edges and nonedges, or as the book says, it preserves adjacency and nonadjacency. Application of fuzzy ifthen rule in fuzzy petersen graph. This paper deals with a,bconsecutive labelings of generalized petersen graphpn,k. If every vertex has degree at least n 2, then g has a hamiltonian cycle. E, where v is a nite set and graph, g e v 2 is a set of pairs of elements in v. Girth in this section we will establish the exact value of the girth of a generalized petersen graph gpn. Total domination number of generalized petersen graphs pck,k. In order to state the main result we need to introduce the graph and two 2 parametric families of cubic, connected graphs. In chapter 3, we determine the structure of any large internally 4connected graph which has no p 2 minor, where p 2 is a graph on 8 vertices, edges, and is isomorphic to the graph created by contracting two edges of a perfect matching of the petersen graph. If e 0, the graph consists of a single node with a single face surrounding it. Electronic notes in discrete mathematics 48 2015 235a241236 2 a class of super edge magic.
The petersen graph is 3edgecolorable if and only if its edgeset can be expressed as a union of three disjoint perfect matchings, say m 1. The dodecahedron can be expressed as a graph shown in figure 1. I am trying to find out the automorphism group of the petersen graph. We prove the exact formulae for gpn, 1 and gpn, 2, while for other values of k a lower bound is stated. Strong matching preclusion of generalized petersen graphs. Ten things you possibly didnt know about the petersen graph. These graphs were introduced by coxeter 1950 and named by watkins 1969. Disproof assume, for contradiction, that the partition.
In, watkins introduced the notion of generalized petersen graph. Prove petersen graph is not hamiltonian using deduction and. The petersen graph is not 3edgecolorablea new proof. Super edge magic graceful labeling of generalized petersen. Generalizing the generalized petersen graphs sciencedirect. All other generalized petersen graph has chromatic index 3. G is the order of a largest set of independent vertices. The generalized petersen graph pn, k has vertices, and respectively, edges given by.
The spectrum of generalized petersen graphs the australasian. Petersen introduced the most well known graph, the petersen graph, as an example of a cubic bridgeless graph that is not tait colorable, i. In this account, the authors examine those areas, using the prominent role of the petersen graph as a unifying feature. The petersen graph is a small graph that serves as a useful example and counterexample in graph theory. Graphs and matrices 1 the adjacency matrix of a graph. By contrast, we study an infinite family of regular graphs, the generalized petersen graphs g n. The smallest eigenvalue gives information about independence number and chromatic number. The generalized petersen graph gpg pn, k has vertices, respectively, edges. Sep 06, 20 however, removing any vertex leaves a hamiltonian graph, so the petersen graph is described as hypohamiltonian. The petersen graph is most commonly drawn as a pentagram inside with five spokes. Jan 29, 2016 petersen graph in the mathematical field of graph theory, the petersen graph is an undirected graph with 10 vertices and 15 edges. The set v is called the set of vertices and eis called the set of edges of g. The groups of the generalized petersen graphs cambridge core.
The petersen graph occupies an important position in the development of several areas of modern graph theory, because it often appears as a counterexample to important conjectures. Hyperhamiltonian generalized petersen graphs request pdf. Arora, novi high school the strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almostperfect matchings. The edge metric dimension problem was recently introduced, which initiated the study of its mathematical properties. M 2 and m 3 the petersen graph consists of two cycles of length 5, which we denote c 1 and c 2, joined by a set of ve. Theorem dirac let g be a simple graph with n 3 vertices. On minimum vertex covers of generalized petersen graphs. In this paper, we have developed an algorithm to find the minimal total dominating set of the generalized petersen graphs, when. Characterization of generalized petersen graphs that are. Petersen graph in the mathematical field of graph theory, the petersen graph is an undirected graph with 10 vertices and 15 edges.
Since the generalized petersen graph is cubic, mn32, where m is the edge. While all regular petersen graphs admit game con gurations that are not solvable, we prove. There are exactly connected cubic graphs on vertices. Component connectivity of generalized petersen graphs. The petersen graph is the smallest 3copwin graph andrew beveridge. Indeed, the petersen graph is the smallest hypohamiltonian graph. Peterson graph is a 3regular graph ion 4 incidence matrix, of a graph g with n vertices m edges and without selfloops is an n x m matrix a aij. It is wellknown that the vertexcover problem is an npcomplete problem 6, p.
A total dominating set of a graph g is a set of the vertex set v of g such that every vertex of g is adjacent to a vertex in s. Edge metric dimension of some generalized petersen graphs. Pdf we completely describe the spectrum of the generalized petersen. The groups of the generalized petersen graphs mathematical. Graph domination numbers and algorithms for finding them have been investigated for numerous classes of graphs, usually for graphs that have some kind of treelike structure. The petersen graph is the complement of the johnson graph j5,2. In general, the more edges a graph has, the more likely it is to have a hamiltonian cycle. The petersen graph is a graph with 10 vertices and 15 edges. That is, it is a unit distance graph the simplest nonorientable surface on which the petersen graph can be embedded without crossings is the projective plane. Wilson conjectured that, if the generalized petersen graph gpn, k is non triv ially unstable, then both n a nd k are even, and either n 2 is o dd and k 2. The generalized petersen graph gn,k is formed by connecting the vertices of a regular ngon to the corresponding. Sheikholeslami, 2 pu wu, 1,3 and jiabiao liu 4 1 ins tit ute of c omp utin g scien ce and t e chno logy, gua ngzh ou. Labeled petersen graph with new tikz graph library tex.
The induction takes the form of mapping a hamilton path in the smaller graph piecewise to the larger graph an inserting subpaths we call rotors to obtain a hamilton path in the larger graph. Given that it is a graph on 10 vertices with each vertex being adjacent to 3 edges, it is possible to ask the question whether the complete graph on 10 vertices can be decomposed into 3 copies of the petersen graph. For each vertex v there are t vertices u so that uv 2eg and vu 2eg. The generalized petersen graphs gpgs which have been invented by watkins, may serve for perhaps the. It is a small graph that serves as a useful example and. Can we partition the edges of 10 into three disjoint sets, such that each set forms a petersen graph. In this case, the dual graph is cubic and bridgeless, so by petersen s theorem it has a matching, which corresponds in the original graph to a pairing of adjacent triangle faces. Thus, no cycle of length 4 lies in the petersen graph.
Lights out on petersen graphs bruce torrence and robert torrence august 30, 20 abstract we establish some preliminary results for sutners. A vertex cover with minimum size in a graph g is called a minimum vertex cover of g and its size is denoted by. You can verify this last part for yourself and it is a simple combinatorial argument. Strong matching preclusion of generalized petersen graphs e.
It is easy to see the equivalence between the following statements, but most proofs for each of them use a case by case argument 1. The petersen graph can also be drawn with crossings in the plane in such a way that all the edges have equal length. Pdf the spectrum of generalized petersen graphs researchgate. The metric dimension of some generalized petersen graphs zehui shao, 1 s. The superconnectivity superedgeconnectivity of a connected graph g is the minimum number of vertices edges that need to be deleted from g in order to disconnect g without creating isolated vertices. The theoretical properties of the edge metric representations and the edge metric dimension of generalized petersen graphs gpn, k are studied in this paper.
We present two algorithms which between them lead to the determination of upper and lower bounds on the domination numbers of these graphs, and we believe that our upper bound may be the exact value. The independence number for the generalized petersen graphs. Petersen graph, being a snark, has a chromatic index of 4. A set q of vertices of a graph g v,e is called a vertex cover, if each edge in e has at least one endpoint in q. Total domination number of generalized petersen graphs. Pdf on the metric dimension of generalized petersen graphs. We now state our main theorem which adds to the class of graphs whose spectrum. Pdf the metric dimension of some generalized petersen graphs. A n,k,t,l,m directed strongly regular graph is a directed graph g on n vertices, each of in and outdegree k, with the conditions if uv 2eg, then there are l vertices x so that ux 2eg and xv 2eg. The spectrum of generalized petersen graphs 41 of a cycle graph on n vertices c n, respectively, cn k is the union of d cycle graphs c nd on nd vertices, where d gcdn,k.
Petersen s theorem can also be applied to show that every maximal planar graph can be decomposed into a set of edgedisjoint paths of length three. In this research, we determine partitiondimension of caveman graphcn. In 10 every vertex is of degree 9, and in the petersen graph 3. The petersen graph has a subdivision which is k5 contract edges connecting the 5 sided star to the edge.
The generalized petersen graph g9,2 is one of the few graphs known to have only one 3edgecoloring. Partition dimension, caveman graph, generalized petersen graph,km. We now state our main theorem which adds to the class of graphs whose. Petersens theorem can also be applied to show that every maximal planar graph can be decomposed into a set of edgedisjoint paths of length three.
Julius petersen mathematician for the mathematician after whom this graph is named. On the cop turn, each cop may move to a neighboring vertex or remain in place. The outer subgraph whose adjacency matrix is cnofpn,k is the cycle graph c n and the inner subgraph whose adjacency matrix is cn khasd connected components. Kuratowskis theorem tells us that if k5 is a subdivision of a graph then that graph is not planar. Introductionthere are many concepts in graph discussed by researchers. A regular graph of degree 3 is called a cubic graph of a has exactly two one.
The second largest eigenvalue of a graph gives information about expansion and randomness properties. Some progress was made by meszaros in 2008 21 when the petersen graph was found to be 4ordered and the heawood graph was proved to be 4orderedhamiltonian. Deduce that the petersen graph does not contain a hamiltonian cycle. It is the unique strongly regular graph with parameters v 10, k 3. Pdf on some aspects of the generalized petersen graph.
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