Petersen Graph Research Articles

Some Results on Prime and Distinct Prime Distance Labeling of Graphs

Let 𝒁 be the set of all integers. A graph 𝑯 is a prime distance graph if there exists an injective function 𝒈:𝑽(𝑯)→𝒁 such that for any two adjacent vertices 𝒙 and 𝒚, the integer |𝒈(𝒙)−𝒈(𝒚)| is a prime. So 𝑯 is a prime distance graph if and only if there exists a prime distance labeling of 𝑯. If the edge labels of 𝑯 are also distinct, then 𝒈 is called a distinct prime distance labeling of 𝑯 and 𝑯 is called a distinct prime distance graph. The generalized Petersen graphs 𝑷(𝒏,𝒌) are defined to be a graph on 𝟐𝒏(𝒏≥𝟑) vertices with 𝑽(𝑷(𝒏,𝒌))={𝒗𝒊,𝒖𝒊:𝟎≤𝒊≤𝒏−𝟏}and 𝑬(𝑷(𝒏,𝒌))={𝒗𝒊𝒗𝒊+𝟏,𝒗𝒊𝒖𝒊,𝒖𝒊𝒖𝒊+𝒌:𝟎≤𝒊≤𝒏−𝟏, subscripts modulo 𝒏}. In this paper, we show that the generalized Petersen graphs 𝑷(𝒏,𝟑) permit a prime distance labeling for all even 𝒏>𝟓 and conjecture that 𝑷(𝒏,𝟐) and 𝑷(𝒏,𝟑) admit a prime distance labeling for any 𝒏≥𝟓 and all odd 𝒏≥𝟓, respectively. We also prove that the cycle 𝑪𝒏 admits a distinct prime distance labeling for all 𝒏≥𝟑, besides establishing the prime distance labeling for some graphs.

On the Local Adjacency Metric Dimension of Generalized Petersen Graphs

The local adjacency metric dimension is one of graph topic. Suppose there are three neighboring vertex , , in path . Path is called local if where each has representation: a is not equals and may equals to . Let’s say, . For an order set of vertices , the adjacency representation of with respect to is the ordered -tuple , where represents the adjacency distance . The distance defined by 0 if , 1 if adjacent with , and 2 if does not adjacent with . The set is a local adjacency resolving set of if for every two distinct vertices , and adjacent with y then . A minimum local adjacency resolving set in is called local adjacency metric basis. The cardinality of vertices in the basis is a local adjacency metric dimension of , denoted by . Next, we investigate the local adjacency metric dimension of generalized petersen graph.

The number of switching isomorphism classes of signed graphs associated with particular graphs

Two signed graphs are called switching isomorphic if one of them is isomorphic to a switching equivalent of the other. To determine the number of switching non-isomorphic signed graphs on a specific graph, we will establish a method based on the action of its automorphism group. As an application and computational result, we classify all the switching non-isomorphic signed graphs arising from the complete graph K5 and the generalized Petersen graph GP(7,2). The instances have minimum possible negative edges between switching equivalency classes. Moreover, some results on the automorphism groups of the target signed graphs are obtained.

OTSMO Labeling on Generalized Petersen Graphs GP(n, 2)

In this paper a discussion of OTSMO Labeling (One Third Sum Modulo 3, introduced by the Researcher) on Generalized Petersen graphs GP(n, 2) is done.

Girth Six Cubic Graphs Have Petersen Minors

We prove that every 3-regular graph with no circuit of length less than six has a subgraph isomorphic to a subdivision of the Petersen graph.

The smallest nontrivial snarks of oddness 4

The oddness of a cubic graph is the smallest number of odd circuits in a 2-factor of the graph. This invariant is widely considered to be one of the most important measures of uncolourability of cubic graphs and as such has been repeatedly reoccurring in numerous investigations of problems and conjectures surrounding snarks (connected cubic graphs admitting no proper 3-edge-colouring). In Goedgebeur et al., 2019 we have proved that the smallest number of vertices of a snark with cyclic connectivity 4 and oddness 4 is 44. We now show that there are exactly 31 such snarks, all of them having girth 5. These snarks are built up from subgraphs of the Petersen graph and a small number of additional vertices. Depending on their structure they fall into six classes, each class giving rise to an infinite family of snarks with oddness at least 4 with increasing order. We explain the reasons why these snarks have oddness 4 and prove that the 31 snarks form the complete set of snarks with cyclic connectivity 4 and oddness 4 on 44 vertices. The proof is a combination of a purely theoretical approach with extensive computations performed by a computer.

Edge Metric Dimension of Some Generalized Petersen Graphs

The edge metric dimension problem was recently introduced, which initiated the study of its mathematical properties. The theoretical properties of the edge metric representations and the edge metric dimension of generalized Petersen graphs $GP(n,k)$ are studied in this paper. We prove the exact formulae for $GP(n,1)$ and $GP(n, 2)$, while for the other values of $k$ the lower bound is stated.

A note on edge-distance-balanced property of the generalized Petersen graphs GP(4t, 2)

A graph G is said to be edge-distance-balanced if for any edge uv of G, the number of edges closer to u than to v is equal to the number of edges closer to v than to u. Let GP(n, k) be a generalized Petersen graph. It is proven that for any integers t ≥ 5, the generalized Petersen graph GP(4t, 2) is not edge-distance-balanced.

1,2]-Domination in generalized Petersen graphs

A vertex subset [Formula: see text] of a graph [Formula: see text] is a [Formula: see text]-dominating set if each vertex of [Formula: see text] is adjacent to either one or two vertices in [Formula: see text]. The minimum cardinality of a [Formula: see text]-dominating set of [Formula: see text], denoted by [Formula: see text], is called the [Formula: see text]-domination number of [Formula: see text]. In this paper, the [Formula: see text]-domination and the [Formula: see text]-total domination numbers of the generalized Petersen graphs [Formula: see text] are determined.

The Locating-Chromatic Number for Certain Operation of Generalized Petersen Graphs sP(4,2)

The locating-chromatic number of a graph combined two graph concept, coloring vertices and partition dimension of a graph. The locating-chromatic number, denoted by χL(G), is the smallest k such that G has a locating k-coloring. In this paper, we discuss the locating-chromatic number for certain operation of generalized Petersen graphs sP(4,2).

The Italian Domination Numbers of Generalized Petersen Graphs P(n,3)

An Italian dominating function of G is a function f : V ( G ) → { 0 , 1 , 2 } , for every vertex v such that f ( v ) = 0 , it holds that ∑ u ∈ N ( v ) f ( u ) ≥ 2 . The Italian domination number γ I ( G ) is the minimum weight of an Italian dominating function on G. In this paper, we determine the exact values of the Italian domination numbers of P ( n , 3 ) .

Highly-connected planar cubic graphs with few or many Hamilton cycles

In this paper we consider the number of Hamilton cycles in planar cubic graphs of high cyclic edge-connectivity, answering two questions raised by Chia and Thomassen (2012) about extremal graphs in these families. In particular, we find families of cyclically 5-edge-connected planar cubic graphs with more Hamilton cycles than the generalized Petersen graphs P(2n,2). The graphs themselves are fullerene graphs that correspond to certain carbon molecules known as nanotubes—more precisely, the family consists of the zigzag nanotubes of (fixed) width 5and increasing length. In order to count the Hamilton cycles in the nanotubes, we develop methods inspired by the transfer matrices of statistical physics. We outline how these methods can be adapted to count the Hamilton cycles in nanotubes of greater (but still fixed) width, with the caveat that the resulting expressions involve matrix powers. We also consider cyclically 4-edge-connected planar cubic graphs with few Hamilton cycles, and exhibit an infinite family of such graphs each with exactly 4 Hamilton cycles. Finally we consider the “other extreme” for these two classes of graphs, thus investigating cyclically 4-edge-connected planar cubic graphs with many Hamilton cycles and the cyclically 5-edge-connected planar cubic graphs with few Hamilton cycles. In each of these cases, we present partial results, examples and conjectures regarding the graphs with few or many Hamilton cycles.

Maximal labeling of graphs

The notion of maximal labeling, optimal maximal labeling and the maximal index of a graph using the nonunit elements of a commutative ring with identity are introduced and studied. The maximal index of complete graphs, complete bipartite graphs are given. Maximal index of cycles of order up to [Formula: see text] and Petersen graph are also given.

Zk-Magic Labeling of Path Union of Graphs

For any non-trivial Abelian group A under addition a graph G is said to be A-magic if there exists a labeling f : E(G) → A − {0} such that, the vertex labeling f+ defined as f+(v) = ∑f(uv) taken over all edges uv incident at v is a constant. An A-magic graph G is said to be Zk-magic graph if the group A is Zk, the group of integers modulo k and these graphs are referred as k-magic graphs. In this paper we prove that the graphs such as path union of cycle, generalized Petersen graph, shell, wheel, closed helm, double wheel, flower, cylinder, total graph of a path, lotus inside a circle and n-pan graph are Zk-magic graphs.

Long Cycles in t-Tough Graphs with t > 1

It is proved that if G is a t-tough graph of order n and minimum degree δ with t > 1, then either G has a cycle of length at least min{n, 2δ + 4} or G is the Petersen graphIt is proved that if G is a t-tough graph of order n and minimum degree δ with t > 1, then either G has a cycle of length at least min{n, 2δ + 4} or G is the Petersen graph

Complexity of discrete Seifert foliations over a graph

In the present paper, we study the complexity of an infinite family of graphs Hn = Hn(G1, G2, ..., Gm) that are discrete Seifert foliations over a graph H on m vertices with fibers G1, G2, ..., Gm. Each fiber Gi = Cn(si,1, si,2, ..., si,ki) of this foliation is the circulant graph on n vertices with jumps si,1, si,2, ..., si,ki. The family of discrete Seifert foliations is sufficiently large. It includes the generalized Petersen graphs, I-graphs, Y-graphs, H-graphs, sandwiches of circulant graphs, discrete torus graph and others. We obtain a closed formula for the number t(n) of spanning trees in Hn in terms of Chebyshev polynomials, investigate some arithmetical properties of this function and find its asymptotics as n → ∞.

Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

AbstractA ‐bisection of a bridgeless cubic graph is a ‐colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes ( monochromatic components in what follows) have order at most . Ban and Linial Conjectured that every bridgeless cubic graph admits a ‐bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph with has a ‐edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (ie, a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we provide evidence of a strong relation of the conjectures of Ban‐Linial and Wormald with Ando's Conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above‐mentioned conjectures. Moreover, we prove Ban‐Linial's Conjecture for cubic‐cycle permutation graphs. As a by‐product of studying ‐edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.

Petersen type n-circulant and weak metacirculant

In this paper, we extend the class of multilayer generalized Petersen graphs introduced in [J. Combin. Theory Ser. A 155 (2018) 225–243] to a class of Petersen type n-circulants. We give a characterization of G-vertex-transitive Petersen type n-circulants Γ of odd prime power order and smallest possible valency, where G ≤ Aut (Γ) is a metacyclic group. As a result, we construct a class of non-Cayley graphs which have a vertex-transitive non-split metacyclic group of automorphisms. This corrects an error in the literature regarding weak metacirculants.

Coloring of Strong Intuitionistic Fuzzy Petersen Graph PG

Coloring of Strong Intuitionistic Fuzzy Petersen Graph PG

On Certain Edge-Transitive Bicirculants

A graph $\Gamma$ of even order is a bicirculant if it admits an automorphism with two orbits of equal length. Symmetry properties of bicirculants, for which at least one of the induced subgraphs on the two orbits of the corresponding semiregular automorphism is a cycle, have been studied, at least for the few smallest possible valences. For valences $3$, $4$ and $5$, where the corresponding bicirculants are called generalized Petersen graphs, Rose window graphs and Tabač jn graphs, respectively, all edge-transitive members have been classified. While there are only 7 edge-transitive generalized Petersen graphs and only 3 edge-transitive Tabač jn graphs, infinite families of edge-transitive Rose window graphs exist. The main theme of this paper is the question of the existence of such bicirculants for higher valences. It is proved that infinite families of edge-transitive examples of valence $6$ exist and among them infinitely many arc-transitive as well as infinitely many half-arc-transitive members are identified. Moreover, the classification of the ones of valence $6$ and girth $3$ is given. As a corollary, an infinite family of half-arc-transitive graphs of valence $6$ with universal reachability relation, which were thus far not known to exist, is obtained.