The petersen graph is a graph with 10 vertices and 15 edges. In this paper we present a short survey of the status of this conjecture and. The edge metric dimension problem was recently introduced, which initiated the study of its mathematical properties. On the prime labeling of generalized petersen graphs pn,3 1 kh. Hamiltonian decomposition of prisms over cubic graphs discrete. Extremal combinatorics in generalized kneser graphs pure. It is proved that the generalized petersen graph pn,k is hypohamiltonian if and only if k2 and n. They include the petersen graph and generalize one of the ways of constructing the petersen graph. Pdf on some aspects of the generalized petersen graph. While obtaining some interesting results, we also provide relevant background and problems. The pdf file you selected should load here if your web browser has a pdf reader plugin installed for example, a recent version of adobe acrobat reader if you would like more information about how to print, save, and work with pdfs, highwire press provides a helpful frequently asked questions about pdfs alternatively, you can download the pdf. The classification of hamiltonian generalized petersen graphs core. In this paper, we first extend the results of the metric dimension of and and study bounds on the metric dimension of the families of the generalized petersen graphs and.
This page was last edited on 31 august 2019, at 14. In this paper, we have developed an algorithm to find the minimal total dominating set of the generalized petersen graphs. The generalized petersen graph gn,k is formed by connecting the vertices of a regular ngon to the. The metric dimension of some generalized petersen graphs. Component connectivity of generalized petersen graphs. Girth in this section we will establish the exact value of the girth of a generalized petersen graph.
Mathematics free fulltext edge irregular reflexive. The independence number for the generalized petersen graphs. On strong rainbow vertexcoloring of generalized petersen graphs 1001 proof. The diameter vulnerability of the generalized petersen. The generalized petersen graph is a famous class of graphs in graph theory. Sharply transitive decompositions of complete graphs into generalized petersen graphs. There are only few results concerning the crossing numbers of join of some graphs. Abstract by identifying a local property which structurally classi.
The crossing number of join of the generalized petersen. We characterize generalized petersen graph which have efficient signed roman domination number. The tutte polynomial and the generalized petersen graph. Petersen graph with new tikz graph library tex latex. Recognizing generalized petersen graphs in linear time.
We characterize all the members of generalized petersen graphs that are kronecker covers, and describe the structure of their respective quotients. In 1, we introduced evidence labeling and showed the admissibility of evidence labeling in path related graphs. We show that the generalized petersen graph pm,2 and the line graph of pm,2 are tunique. Media in category generalized petersen graphs the following 5 files are in this category, out of 5 total. The proof of i is based on the classification of orientable regular embeddings of the n. 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. A graph with such a capacity characterized is known as a vertexlabeled graph.
Generalized petersen graphs are an important class of commonly used interconnection networks and have been studied. In this paper, the crossing numbers of join products for the generalized petersen graph p3, 1 with n isolated. In the mathematical field of graph theory, the petersen graph is an undirected graph with 10. In the mathematical field of graph theory, the petersen graph is an undirected graph with 10 vertices and 15 edges. Edge metric dimension of some generalized petersen graphs. Generalized petersen graphs brian alspach department of mathematics, simon fraser university, burnaby. Our result in the next theorem we have proved that the minimum cardinality of the oldset of generalized petersen graph. In this paper, we study the independence number for the generalized petersen graphs. Figure 3 shows that the petersen graph is prismdecomposable and also. Therefore, it is interesting to characterize the double roman graphs in generalized petersen graphs. On the local adjacency metric dimension of generalized. The petersen graph is named after julius petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no threeedgecoloring. Hyper hamiltonian generalized petersen graphs tacheng maia, jengjung wanga.
A graph labeling is the task of integers, generally spoken to by whole. Formally, given a graph g v, e a vertex labeling is a capacity from v to an arrangement of integers. In 2008 wilson conjectured that, if the generalized petersen graph gpn,k is nontrivially unstable, then both n and k are even, and either n2 is odd and. We consider the class of i graphs, which is a generalization of the class of the generalized petersen graphs. The classification of hamiltonian generalized petersen graphs. The nauru graph can also be constructed as the generalized petersen graph g12, 5 which is formed by the vertices of a dodecagon connected to the vertices of a twelvepoint star in which each point of the.
We show that two i graphs in, j, k and in, j 1, k 1 are isomorphic if and only if there. 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. Pdf on the metric dimension of generalized petersen. Generalizing the generalized petersen graphs discrete. Chemical indices of generalized petersen graph yuan liu, liancui zuoy and chunhong shang abstract the generalized petersen graph gpn. Available formats pdf please select a format to send. Pdf the spectrum of generalized petersen graphs researchgate.
The generalized petersen graphs gpgs which have been invented by watkins, may serve for perhaps the simplest nontrivial examples of galactic graphs, i. Injective chromatic number of outerplanar graphs mozafarinia, mahsa and omoomi, behnaz, taiwanese journal of mathematics, 2018. 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 each isomorphic to c nd. Publishers pdf, also known as version of record includes final page, issue and volume. On the domination number of the generalized petersen graphs. The total domination number of generalized petersen graphs pm,2 is obtained in. Some of them are also vertextransitive or even more highly symmetric, and some are cayley graphs.
Edge irregular reflexive labeling for disjoint union of generalized. On strong rainbow vertexcoloring of generalized petersen. 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. The groups of the generalized petersen graphs volume 70 issue 2 roberto frucht, jack e. We have published a related paper entitled on the bondage number of a graph in discrete mathematics. Total domination number of generalized petersen graphs. The 2rainbow bondage number in generalized petersen graphs. Signed roman domination number of generalized petersen.
In this paper, we investigate the 2rainbow bondage problem on generalized petersen graphs. Pdf we completely describe the spectrum of the generalized petersen graph p n,k, thus adding to the classes of graphs whose spectrum is known. In conclusion, we examine how fruchts theorem applies to the speci c case of cubic hamiltonian graphs. Which generalized petersen graphs are cayley graphs. We have published a related paper entitled on the bondage number of a graph. All structured data from the file and property namespaces is available under the. The theoretical properties of the edge metric representations and the edge metric dimension of generalized petersen graphs. The cardinality of vertices in the basis is a local adjacency metric dimension of, denoted by.
G is the order of a largest set of independent vertices. The obtained results mean that these families of graphs have constant metric dimension. The class of generalized petersen graphs was introduced by coxeter 6 in 1950 and its name was coined in 1969 by watkins. Abstract the generalized petersen graph gp n, k, n. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. Vol 5, no 2 2017 electronic journal of graph theory. Pdf characterization of generalized petersen graphs that. Edge magic, super edge magic, super magic, petersen graph.
A minimum local adjacency resolving set in is called local adjacency metric basis. In this paper, we will give some necessary conditions and some sufficient conditions for the hyperhamiltonian generalized petersen graphs. A graph labeling is the task of integers, generally spoken to by whole numbers, to the edges or vertices, or both of a graph. Wilson2 1university of primorska, slovenia, and university of salzburg, austria. Lights out on petersen graphs bruce torrence and robert torrence august 30, 20 abstract we establish some preliminary results for sutners. Watkins the groups of the generalized petersen graphs. Im specifically interested in a way to avoid having to define a new counter. As an interesting family of graphs, the domination and its variations of generalized petersen graphs have attracted considerable attention 1,16. By contrast, we study an infinite family of regular graphs, the generalized petersen graphs.
We observe that some of such quotients are again generalized petersen graphs. Recognizing generalized petersen graphs in linear time matjaz krnc1 and robin j. In graph theory, the generalized petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. 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. G is tunique if any arbitrary graph h being tequivalent to g implies that h is isomorphic to g.
Mominul haque department of computer science and engineering shahjalal university of science and technology sylhet3114. Theorem, followed by an indepth examination of the automorphism groups of generalized petersen graphs and cubic hamiltonian graphs in lcf notation. Next, we investigate the local adjacency metric dimension of generalized petersen graph. Files are available under licenses specified on their description page. In this paper we show the admissibility of evidence labeling in some generalized petersen graphs. Simmons, a status report on the cpa conjecture, congres. Im considering moving from tkzberge to the new tikz graph library for drawing my graphs in the sense of graph theory.
Game chromatic number of generalized petersen graphs and. In this paper, we obtain kth signed spectral moment and kth signed laplacian. Characterization of generalized petersen graphs that are kronecker. On the prime labeling of generalized petersen graphs p n. Characterization of generalized petersen graphs that are. In this paper we probe its spectrum and determine the estrada index, laplacian estrada index, signless laplacian estrada index, normalized laplacian estrada index, and energy of a graph. British columbia wa is6, canada communicated by the editors received february 26, 1982 the generalized petersen graph. Since n is even, the outer rim is an even cycle of length and there are two inner rims each of which is a cycle of length 2 n.