The crossing number of join of the generalized petersen. In this paper we present a short survey of the status of this conjecture and. Characterization of generalized petersen graphs that are. In this paper, we have developed an algorithm to find the minimal total dominating set of the 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 each isomorphic to c nd. G is the order of a largest set of independent vertices. Recognizing generalized petersen graphs in linear time matjaz krnc1 and robin j. British columbia wa is6, canada communicated by the editors received february 26, 1982 the generalized petersen graph. Game chromatic number of generalized petersen graphs and. In this paper, we obtain kth signed spectral moment and kth signed laplacian.
On the prime labeling of generalized petersen graphs pn,3 1 kh. We characterize generalized petersen graph which have efficient signed roman domination number. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. In this paper, we will give some necessary conditions and some sufficient conditions for the hyperhamiltonian generalized petersen graphs. 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.
Mathematics free fulltext edge irregular reflexive. The independence number for the generalized petersen graphs. We have published a related paper entitled on the bondage number of a graph. 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. Which generalized petersen graphs are cayley graphs. Hyper hamiltonian generalized petersen graphs tacheng maia, jengjung wanga. On the prime labeling of generalized petersen graphs p n. In this paper, we study the independence number for the generalized petersen graphs. The diameter vulnerability of the generalized petersen. The classification of hamiltonian generalized petersen graphs core. In this paper we show the admissibility of evidence labeling in some generalized petersen graphs. A graph with such a capacity characterized is known as a vertexlabeled graph.
The metric dimension of some generalized petersen graphs. G is tunique if any arbitrary graph h being tequivalent to g implies that h is isomorphic to g. The classification of hamiltonian generalized petersen graphs. On the domination number of the generalized petersen graphs.
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. It is proved that the generalized petersen graph pn,k is hypohamiltonian if and only if k2 and n. Recognizing generalized petersen graphs in linear time. We have published a related paper entitled on the bondage number of a graph in discrete mathematics. In conclusion, we examine how fruchts theorem applies to the speci c case of cubic hamiltonian graphs. In this paper, we investigate the 2rainbow bondage problem on generalized petersen graphs. The petersen graph is named after julius petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no threeedgecoloring. Signed roman domination number of generalized petersen. Files are available under licenses specified on their description page. Generalized petersen graphs are an important class of commonly used interconnection networks and have been studied.
We characterize all the members of generalized petersen graphs that are kronecker covers, and describe the structure of their respective quotients. Chemical indices of generalized petersen graph yuan liu, liancui zuoy and chunhong shang abstract the generalized petersen graph gpn. Hamiltonian decomposition of prisms over cubic graphs discrete. Edge metric dimension of some generalized petersen graphs. In the mathematical field of graph theory, the petersen graph is an undirected graph with 10. Publishers pdf, also known as version of record includes final page, issue and volume. Pdf characterization of generalized petersen graphs that. The cardinality of vertices in the basis is a local adjacency metric dimension of, denoted by. 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.
The petersen graph is a graph with 10 vertices and 15 edges. The generalized petersen graph is a famous class of graphs in graph theory. Lights out on petersen graphs bruce torrence and robert torrence august 30, 20 abstract we establish some preliminary results for sutners. Component connectivity of generalized petersen graphs.
This page was last edited on 31 august 2019, at 14. There are only few results concerning the crossing numbers of join of some graphs. By contrast, we study an infinite family of regular graphs, the generalized petersen graphs. In 1, we introduced evidence labeling and showed the admissibility of evidence labeling in path related graphs. The class of generalized petersen graphs was introduced by coxeter 6 in 1950 and its name was coined in 1969 by watkins.
Im considering moving from tkzberge to the new tikz graph library for drawing my graphs in the sense of graph theory. In this paper, the crossing numbers of join products for the generalized petersen graph p3, 1 with n isolated. Simmons, a status report on the cpa conjecture, congres. The proof of i is based on the classification of orientable regular embeddings of the n. We show that the generalized petersen graph pm,2 and the line graph of pm,2 are tunique. A minimum local adjacency resolving set in is called local adjacency metric basis. Therefore, it is interesting to characterize the double roman graphs in generalized petersen graphs. Edge magic, super edge magic, super magic, petersen graph. Abstract by identifying a local property which structurally classi. Media in category generalized petersen graphs the following 5 files are in this category, out of 5 total. While obtaining some interesting results, we also provide relevant background and problems. Watkins the groups of the generalized petersen graphs.
Petersen graph with new tikz graph library tex latex. 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. Vol 5, no 2 2017 electronic journal of graph theory. 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. 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. Edge irregular reflexive labeling for disjoint union of generalized. 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. As an interesting family of graphs, the domination and its variations of generalized petersen graphs have attracted considerable attention 1,16.
Theorem, followed by an indepth examination of the automorphism groups of generalized petersen graphs and cubic hamiltonian graphs in lcf notation. Figure 3 shows that the petersen graph is prismdecomposable and also. All structured data from the file and property namespaces is available under the. Pdf on some aspects of the generalized petersen graph. The tutte polynomial and the generalized petersen graph. The generalized petersen graphs gpgs which have been invented by watkins, may serve for perhaps the simplest nontrivial examples of galactic graphs, i. 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. Mominul haque department of computer science and engineering shahjalal university of science and technology sylhet3114. They include the petersen graph and generalize one of the ways of constructing the petersen graph.
Im specifically interested in a way to avoid having to define a new counter. A graph labeling is the task of integers, generally spoken to by whole numbers, to the edges or vertices, or both of a graph. The total domination number of generalized petersen graphs pm,2 is obtained in. Injective chromatic number of outerplanar graphs mozafarinia, mahsa and omoomi, behnaz, taiwanese journal of mathematics, 2018. Next, we investigate the local adjacency metric dimension of generalized petersen graph.
In the mathematical field of graph theory, the petersen graph is an undirected graph with 10 vertices and 15 edges. Our result in the next theorem we have proved that the minimum cardinality of the oldset of generalized petersen graph. Available formats pdf please select a format to send. On strong rainbow vertexcoloring of generalized petersen. The generalized petersen graph gn,k is formed by connecting the vertices of a regular ngon to the. Abstract the generalized petersen graph gp n, k, n. The theoretical properties of the edge metric representations and the edge metric dimension of generalized petersen graphs. Similarly, a hamiltonian graph g is hyperhamiltonian if g. Wilson2 1university of primorska, slovenia, and university of salzburg, austria. A graph labeling is the task of integers, generally spoken to by whole. Some of them are also vertextransitive or even more highly symmetric, and some are cayley graphs.
The 2rainbow bondage number in generalized petersen graphs. Generalized petersen graphs brian alspach department of mathematics, simon fraser university, burnaby. 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 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. Characterization of generalized petersen graphs that are kronecker. Generalizing the generalized petersen graphs discrete.
We show that two i graphs in, j, k and in, j 1, k 1 are isomorphic if and only if there. Total domination number of generalized petersen graphs. Sharply transitive decompositions of complete graphs into generalized petersen graphs. Pdf the spectrum of generalized petersen graphs researchgate. On strong rainbow vertexcoloring of generalized petersen graphs 1001 proof. We observe that some of such quotients are again generalized petersen graphs. We consider the class of i graphs, which is a generalization of the class of the generalized petersen graphs. 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.748 859 1386 1368 302 1441 486 747 1033 1483 233 320 1095 867 694 418 7 351 722 400 700 509 320 1449 544 1206 33 19 196 1229 1295