The set v is called the set of vertices and eis called the set of edges of g. Hyperhamiltonian generalized petersen graphs request pdf. 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. Labeled petersen graph with new tikz graph library tex. Total domination number of generalized petersen graphs pck,k. In the mathematical field of graph theory, the petersen graph is an undirected graph with 10. 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. The dodecahedron can be expressed as a graph shown in figure 1. 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. Automorphisms of graphs math 381 spring 2011 an automorphism of a graph is an isomorphism with itself. The independence number for the generalized petersen graphs. The petersen graph is the complement of the johnson graph j5,2. This paper deals with a,bconsecutive labelings of generalized petersen graphpn,k.
The petersen graph is the smallest 3copwin graph andrew beveridge. While all regular petersen graphs admit game con gurations that are not solvable, we prove. The petersen graph is a graph with 10 vertices and 15 edges. 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. Disproof assume, for contradiction, that the partition. Component connectivity of generalized petersen graphs.
Indeed, the petersen graph is the smallest hypohamiltonian graph. Petersen graph, being a snark, has a chromatic index of 4. Thus, no cycle of length 4 lies in the petersen graph. The groups of the generalized petersen graphs cambridge core. 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. The edge metric dimension problem was recently introduced, which initiated the study of its mathematical properties. In 10 every vertex is of degree 9, and in the petersen graph 3. The spectrum of generalized petersen graphs the australasian. Strong matching preclusion of generalized petersen graphs e. For each vertex v there are t vertices u so that uv 2eg and vu 2eg. G is the order of a largest set of independent vertices. In this research, we determine partitiondimension of caveman graphcn. 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. The petersen graph australian mathematical society lecture.
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. Edge metric dimension of some generalized petersen graphs. 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. Partition dimension, caveman graph, generalized petersen graph,km. The groups of the generalized petersen graphs volume 70 issue 2 roberto frucht, jack e. The groups of the generalized petersen graphs mathematical. We now state our main theorem which adds to the class of graphs whose. Since the generalized petersen graph is cubic, mn32, where m is the edge. You can verify this last part for yourself and it is a simple combinatorial argument. The metric dimension of some generalized petersen graphs zehui shao, 1 s. 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.
On the cop turn, each cop may move to a neighboring vertex or remain in place. The petersen graph has a subdivision which is k5 contract edges connecting the 5 sided star to the edge. Characterization of generalized petersen graphs that are. Generalizing the generalized petersen graphs sciencedirect. Girth in this section we will establish the exact value of the girth of a generalized petersen graph gpn. The generalized petersen graph g9,2 is one of the few graphs known to have only one 3edgecoloring. Kuratowskis theorem tells us that if k5 is a subdivision of a graph then that graph is not planar. On the metric dimension of generalized petersen graphs article pdf available in ars combinatoria waterloo then winnipeg 105.
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. 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. 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. On minimum vertex cover of generalized petersen graphs. 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.
Can we partition the edges of 10 into three disjoint sets, such that each set forms a petersen graph. Petersen introduced the most well known graph, the petersen graph, as an example of a cubic bridgeless graph that is not tait colorable, i. Pdf the metric dimension of some generalized petersen graphs. The petersen graph is the complement of the line graph of. 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.
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 now state our main theorem which adds to the class of graphs whose spectrum. It is wellknown that the vertexcover problem is an npcomplete problem 6, p. The petersen graph does not have a hamiltonian cycle.
Pdf on the metric dimension of generalized petersen graphs. 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. The generalized petersen graph gn,k is formed by connecting the vertices of a regular ngon to the corresponding. Vg, such that u v is an edge if and only if uvis an edge. These graphs were introduced by coxeter 1950 and named by watkins 1969. 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. 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.
Theorem dirac let g be a simple graph with n 3 vertices. 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. Sep 06, 20 however, removing any vertex leaves a hamiltonian graph, so the petersen graph is described as hypohamiltonian. Electronic notes in discrete mathematics 48 2015 235a241236 2 a class of super edge magic. A vertex cover with minimum size in a graph g is called a minimum vertex cover of g and its size is denoted by.
Pdf on some aspects of the generalized petersen graph. Prove petersen graph is not hamiltonian using deduction and. Total domination number of generalized petersen graphs. A solution of the hamiltons game is illustrated by the heavy edges in figure 1. The smallest eigenvalue gives information about independence number and chromatic number. 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. It is the unique strongly regular graph with parameters v 10, k 3. 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. 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. 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. Many additional facts about the petersen graph can be found in 4, 8.
A regular graph of degree 3 is called a cubic graph of a has exactly two one. Introductionthere are many concepts in graph discussed by researchers. The generalized petersen graph gpg pn, k has vertices, respectively, edges. If every vertex has degree at least n 2, then g has a hamiltonian cycle. A borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs pdf. Lights out on petersen graphs bruce torrence and robert torrence august 30, 20 abstract we establish some preliminary results for sutners.
A graph that has fascinated graph theorists over the years because of its appearance as a counterexample in so many areas of the subject. In order to state the main result we need to introduce the graph and two 2 parametric families of cubic, connected graphs. In this paper, we have developed an algorithm to find the minimal total dominating set of the generalized petersen graphs, when. Strong matching preclusion of generalized petersen graphs. The petersen graph can also be drawn with crossings in the plane in such a way that all the edges have equal length. 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.
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. Following up on petersen graph with new tikz graph library, i tried to label the vertices with something different than numbers. Pdf the spectrum of generalized petersen graphs researchgate. There are exactly connected cubic graphs on vertices. E, where v is a nite set and graph, g e v 2 is a set of pairs of elements in v. The petersen graph is a small graph that serves as a useful example and counterexample in graph theory. The petersen graph is most commonly drawn as a pentagram inside with five spokes. If uv 62eg, then there are m vertices x so that ux 2eg and xv 2eg. On the domination number of the generalized petersen graphs. 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. In this account, the authors examine those areas, using the prominent role of the petersen graph as a unifying feature. May, 20 kuratowskis theorem tells us that if k5 is a subdivision of a graph then that graph is not planar. In this case, the dual graph is cubic and bridgeless, so by petersens theorem it has a matching, which corresponds in.
However, removing any vertex leaves a hamiltonian graph, so the petersen graph is described as hypohamiltonian. 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. 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. Ten things you possibly didnt know about the petersen graph. Pdf we completely describe the spectrum of the generalized petersen. 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. 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 petersen graph is 3edgecolorable if and only if its edgeset can be expressed as a union of three disjoint perfect matchings, say m 1. 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. In general, the more edges a graph has, the more likely it is to have a hamiltonian cycle.
The petersen graph g is the simple graph with 10verties and 15edges. Petersen graph in the mathematical field of graph theory, the petersen graph is an undirected graph with 10 vertices and 15 edges. The generalized petersen graph pn, k has vertices, and respectively, edges given by. All other generalized petersen graph has chromatic index 3.
Application of fuzzy ifthen rule in fuzzy petersen graph. In, watkins introduced the notion of generalized petersen graph. 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. If e 0, the graph consists of a single node with a single face surrounding it.
We prove the exact formulae for gpn, 1 and gpn, 2, while for other values of k a lower bound is stated. Deduce that the petersen graph does not contain a hamiltonian cycle. On minimum vertex covers of generalized petersen graphs. The main purpose of this paper is to prove two results about the spectrum of the generalised petersen graphs. We say preserves edges and nonedges, or as the book says, it preserves adjacency and nonadjacency. Julius petersen mathematician for the mathematician after whom this graph is named. Suppose the formula works for all graphs with no more than nedges. On the partition dimension of caveman graph, gneralized. The petersen graph is not 3edgecolorablea new proof. In order to state the main result we need to introduce the graph and two 2parametric families of cubic, connected graphs. In this paper, we prove that the generalized petersen graphs p n, 1,p n, na. Graphs and matrices 1 the adjacency matrix of a graph. Super edge magic graceful labeling of generalized petersen. By contrast, we study an infinite family of regular graphs, the generalized petersen graphs g n.
Let a denote the adjacency matrix of the petersen graph. In this paper, we study the independence number for the generalized petersen graphs. We present two algorithms which between them lead to the determination of upper and lower bounds on the domination numbers of these graphs, and. The number of elements in the set of connected cubic graphs on vertices grows rapidly with. I am trying to find out the automorphism group of the petersen graph. The second largest eigenvalue of a graph gives information about expansion and randomness properties.