The page number is a measure of the quality of a book embedding which is the minimum number of pages in which the graph g can be embedded. Embedding generalized petersen graph in books springerlink. 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. 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. Mominul haque 31 determined the irregular total labelings of generalized petersen graphs. Watkins the groups of the generalized petersen graphs, proceedings of the cambridge.
On strong rainbow vertexcoloring of generalized petersen graphs 1001 proof. Petersen graph, being a snark, has a chromatic index of 4. Abstract by identifying a local property which structurally classi. Generalized petersen graph the generalized petersen graph pn,m has been studied extensively in recent years. On the domination number of the generalized petersen graphs. The isomorphism classes of all generalized petersen graphs pdf. A coloring of g is ddistance if any two vertices at distance at most d from each other get different colors. In this paper, we have developed an algorithm to find the minimal total dominating set of the generalized petersen graphs, when. In conclusion, we examine how fruchts theorem applies to the speci c case of cubic hamiltonian graphs. As an interesting family of graphs, the domination and its variations of generalized petersen graphs have attracted considerable attention 1,16. In this paper, the crossing numbers of join products for the generalized petersen graph p3, 1 with n isolated vertices as well as with the path p n on n vertices and with the cycle c n are determined. Edge metric dimension of some generalized petersen graphs. On strong rainbow vertexcoloring of generalized petersen graphs.
Lj, klabeling number of generalized petersen graph to cite this article. The theoretical properties of the edge metric representations and the edge metric dimension of generalized petersen graphs gpn, k are studied in this paper. Pdf component connectivity of generalized petersen. The obtained results mean that these families of graphs have constant metric dimension. Kronecker cover kc g of a simple undirected graph g is a a special type of bipartite covering graph of g, isomorphic to the direct tensor. In graph theory, the generalized petersen graphs are a family of cubic graphs formed by. Since then this class of graphs has been studied widely because of its interesting traits. 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. Theorem, followed by an indepth examination of the automorphism groups of generalized petersen graphs and cubic hamiltonian graphs in lcf notation.
G is the order of a largest set of independent vertices. British columbia wa is6, canada communicated by the editors received february 26, 1982 the generalized petersen graph gpn, k, n 2 and 1 graphs, notably generalized petersen graphs. In 1950, the generalized petersen graph was introduced by coxeter12 and watkins gave names to these graphs in 1969. If not specified, assume 1 and generate a generalized petersen graph. 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. Component connectivity of generalized petersen graphs. Wilson2 1university of primorska, slovenia, and university of salzburg, austria.
Matching preclusion of the generalized petersen graph by. 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. Our result in the next theorem we have proved that the minimum cardinality of the oldset of generalized petersen graph gpn. On strong rainbow vertexcoloring of generalized petersen. The crossing number of join of the generalized petersen. On some aspects of the generalized petersen graph neliti. Total domination number of generalized petersen graphs pck,k. Mathematics 2018, 6, 304 4 of 10 in this paper, we investigate the re. The petersen graph is named after julius petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no threeedgecoloring. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. As a generalization of the wellknown petersen graph, the generalized petersen graph has attracted much attention.
Generalizing the generalized petersen graphs universita degli. Petersen graph is not a generalized line graph mathematics. The generalized petersen graph gpg pn, k has vertices. Introduction in this paper, g is a simple graph with vertex set v vg. The generalized petersen graph g9,2 is one of the few graphs known to have only one 3edgecoloring. The groups of the generalized petersen graphs volume 70 issue 2 roberto frucht, jack e.
The double roman domination numbers of generalized petersen. Girth in this section we will establish the exact value of the girth of a generalized petersen graph gpn. The classification of hamiltonian generalized petersen graphs core. The diameter vulnerability of the generalized petersen. Multilevel distance labeling for generalized petersen p4k. In, watkins introduced the notion of generalized petersen graph gpg for short as follows. Pdf on the hamilton connectivity of generalized petersen. Paper open access lj, klabeling number of generalized.
Recognizing generalized petersen graphs in linear time. The generalized petersen graph family was introduced in 1950 by h. Narayana prime cordial labeling of generalized petersen graph. In this paper, we investigate the 2rainbow bondage problem on generalized petersen graphs. 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 proof of i is based on the classification of orientable regular embeddings of the n.
Let be a connected graph with vertex set and edge set. Since the generalized petersen graph is cubic, mn32, where m is the edge. We prove the exact formulae for gpn, 1 and gpn, 2, while for other values of k a lower bound is stated. These graphs were introduced by coxeter 1950 and named by watkins 1969. The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion results in a graph with no perfect matchings. The spectrum of generalized petersen graphs 6 proof. The minimum number of colors in ddistance colorings of g is its ddistance chromatic number, denoted by. This graph is a counterexample to many conjectures. Since the three ts can be permuted, we may introduce an extra motion s, of period 3, the generalized petersen graph g24, 5 583 which transforms each tv into the preceding one in cyclic order, so that 2 sts and t3 sts. The spectrum of generalized petersen graphs ralucca gera pantelimon stanic. Now we introduce some basic isomorphisms of the generalized petersen graphs for later reference. The double roman domination numbers of generalized. The spectrum of generalized petersen graphs the australasian.
On minimum vertex covers of generalized petersen graphs. A number of notable 3regular graphs arise as generalized petersen graphs. On the hamilton connectivity of generalized petersen graphs. Chemical indices of generalized petersen graph yuan liu, liancui zuoy and chunhong shang abstract the generalized petersen graph gpn. The class of generalized petersen graphs was introduced by coxeter 6 in 1950 and its name was coined in 1969 by watkins. The generalized petersen graphs gpgs which have been invented by watkins, may serve for perhaps the simplest nontrivial examples of galactic graphs, i. For example papers 1, 3 and 9 are about the hamiltonian character of generalized petersen graphs. The petersen graph is the complement of the line graph of. The independence number for the generalized petersen graphs. Characterization of generalized petersen graphs that are. 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 file directly to your computer, from where it.
Our aim is to find the minimum number of colors required for the u vertices and v vertices. All other generalized petersen graph has chromatic index 3. They include the petersen graph and generalize one of the ways of constructing the petersen graph. The petersen graph p5,2 the matching preclusion problem has been previously studied for some cases of the generalized petersen graphs, all of which are even graphs. The metric dimension of some generalized petersen graphs. We observe that some of such quotients are again generalized petersen graphs, and describe all such pairs. The groups of the generalized petersen graphs mathematical. The 2rainbow bondage number in generalized petersen graphs. The classification of hamiltonian generalized petersen graphs.
Since n is even, the outer rim is an even cycle of length and there are two inner rims each of. The generalized petersen graph g24, 5 sciencedirect. British columbia wa is6, canada communicated by the editors received february 26, 1982 the generalized petersen graph gpn, k, n 2 and 1 graph with an even number of vertices is the minimum number of edges whose deletion results in a graph with no perfect matchings. Available formats pdf please select a format to send. Apr 26, 2016 the page number is a measure of the quality of a book embedding which is the minimum number of pages in which the graph g can be embedded. Therefore, it is interesting to characterize the double roman graphs in generalized petersen graphs.
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 problem was recently introduced, which initiated the study of its mathematical properties. Im currently trying to give a simple argument why the petersen graph is not a generalized line graph it also is an exercise in godsilroyle with following definition. Xu and kang on the power domination number of the generalized petersen graphs, j. Mominul haque department of computer science and engineering. Some of them are also vertextransitive or even more highly symmetric, and some are cayley graphs. In the mathematical field of graph theory, the petersen graph is an undirected graph with 10 vertices and 15 edges.
We characterize generalized petersen graph which have efficient signed roman domination number. Media in category generalized petersen graphs the following 5 files are in this category, out of 5 total. Pdf we completely describe the spectrum of the generalized petersen graph pn,k, thus adding to the classes of graphs whose spectrum is known. The crossing number of join of the generalized petersen graph. In this research we completely describe the spectrum for the class of graphs, defined below. There are papers discussing topics such as tough sets, labeling problems, wide diameters, and coloring of generalized petersen graphs. Generalized petersen graphs brian alspach department of mathematics, simon fraser university, burnaby. 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. Pisanski, tomaz 2010, all generalized petersen graphs are unitdistance graphs pdf, imfm preprints, 1109. We have published a related paper entitled on the bondage number of a graph in discrete mathematics. Which generalized petersen graphs are cayley 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.
On minimum vertex cover of generalized petersen graphs. A vertex subset s of a graph g v,e is a 1, 2dominating set if each vertex of v. 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. Algebraic graph theory is the branch of mathematics that studies graphs by using algebraic properties of associated matrices. Since d 1, then c k is the adjacency matrix of a cycle graph isomorphic to c n, and so it is similar to c, that is, there exists a permutation matrix p, such. On the prime labeling of generalized petersen graphs p n. Domination in the generalized petersen graphs request pdf.
Edge irregular reflexive labeling for disjoint union of. In this paper, we study the independence number for the generalized petersen graphs. More in particular, spectral graph theory studies the relation between graph properties and the spectrum of the adjacency matrix or laplace matrix. Matching preclusion of the generalized petersen graph. Estrada index, and energy of generalized petersen graph pn,m.
Fourth, in any generalized petersen graph a cover which selects all vvertices, besides some uvertices, is called a trivial cover, and it is not a minimum cover. We characterize all the members of generalized petersen graphs that are kronecker covers, and describe the structure of their respective quotients. A graph g is obtained from generalized petersen graph pn,2, where. By contrast, we study an infinite family of regular graphs, the generalized petersen graphs g n. In this paper, the authors discuss the embedding of the generalized petersen graph and determine that the page number of the generalized petersen graph is three in some situations, which is best possible.
Extended in the obvious way to the vertices and edges of the generalized petersen graph gpn,k, t is an automorphism of gpn,k. Recognizing generalized petersen graphs in linear time matjaz krnc1 and robin j. Generalizing the generalized petersen graphs sciencedirect. There are only few results concerning the crossing numbers of join of some graphs. Department of applied mathematics naval postgraduate school monterey, ca 93943 u. Signed roman domination number of generalized petersen graph. Pdf on some aspects of the generalized petersen graph.
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