On cyclic decompositions of the complete graph into the bipartite generalized Petersen graph P(n,3)

2021 ◽  
Vol 344 (5) ◽  
pp. 112339
Author(s):  
Wannasiri Wannasit
Keyword(s):  
Complete Graph ◽  
Petersen Graph ◽  
Generalized Petersen Graph

The Generalized Connectivity of Generalized Petersen Graph

2021 ◽  
Author(s):  
Yuan Si ◽  
Ping Li ◽  
Yuzhi Xiao ◽  
Jinxia Liang
Keyword(s):  
Steiner Trees ◽  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Edge Disjoint ◽  
Generalized Connectivity ◽  
Vertex Set

For a vertex set [Formula: see text] of [Formula: see text], we use [Formula: see text] to denote the maximum number of edge-disjoint Steiner trees of [Formula: see text] such that any two of such trees intersect in [Formula: see text]. The generalized [Formula: see text]-connectivity of [Formula: see text] is defined as [Formula: see text]. We get that for any generalized Petersen graph [Formula: see text] with [Formula: see text], [Formula: see text] when [Formula: see text]. We give the values of [Formula: see text] for Petersen graph [Formula: see text], where [Formula: see text], and the values of [Formula: see text] for generalized Petersen graph [Formula: see text], where [Formula: see text] and [Formula: see text].

The Homology of Abelian Covers of Knotted Graphs

1999 ◽  
Vol 51 (5) ◽  
pp. 1035-1072
Author(s):  
R. A. Litherland
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Petersen Graph ◽  
Infinite Class ◽  
Colored Graphs ◽  
Branched Cover ◽  
Trivalent Graph ◽  
Abelian Covers ◽  
Index 2

AbstractLet be a regular branched cover of a homology 3-sphere M with deck group and branch set a trivalent graph Γ; such a cover is determined by a coloring of the edges of Γ with elements of G. For each index-2 subgroup H of G, MH = /H is a double branched cover of M. Sakuma has proved that H1() is isomorphic, modulo 2-torsion, to ⊕HH1(MH), and has shown that H1() is determined up to isomorphism by ⊕HH1(MH) in certain cases; specifically, when d = 2 and the coloring is such that the branch set of each cover MH → M is connected, and when d = 3 and Γ is the complete graph K4. We prove this for a larger class of coverings: when d = 2, for any coloring of a connected graph; when d = 3 or 4, for an infinite class of colored graphs; and when d = 5, for a single coloring of the Petersen graph.

scholarly journals The K -Size Edge Metric Dimension of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Tanveer Iqbal ◽  
Muhammad Naeem Azhar ◽  
Syed Ahtsham Ul Haq Bokhary
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Complete Bipartite Graph ◽  
Petersen Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Generalized Petersen Graph

In this paper, a new concept k -size edge resolving set for a connected graph G in the context of resolvability of graphs is defined. Some properties and realizable results on k -size edge resolvability of graphs are studied. The existence of this new parameter in different graphs is investigated, and the k -size edge metric dimension of path, cycle, and complete bipartite graph is computed. It is shown that these families have unbounded k -size edge metric dimension. Furthermore, the k-size edge metric dimension of the graphs Pm □ Pn, Pm □ Cn for m, n ≥ 3 and the generalized Petersen graph is determined. It is shown that these families of graphs have constant k -size edge metric dimension.

Total irregularity strength of disjoint union of isomorphic copies of generalized Petersen graph

2017 ◽  
Vol 09 (06) ◽  
pp. 1750071
Author(s):  
Muhammad Naeem ◽  
Muhammad Kamran Siddiqui
Keyword(s):  
Disjoint Union ◽  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Irregularity Strength

Let [Formula: see text] be a graph. A total labeling [Formula: see text] is called totally irregular total [Formula: see text]-labeling of [Formula: see text] if every two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text] satisfy [Formula: see text] and every two distinct edges [Formula: see text] and [Formula: see text] in [Formula: see text] satisfy [Formula: see text] where [Formula: see text] and [Formula: see text] The minimum [Formula: see text] for which a graph [Formula: see text] has a totally irregular total [Formula: see text]-labeling is called the total irregularity strength of [Formula: see text], denoted by [Formula: see text] In this paper, we determined the total irregularity strength of disjoint union of [Formula: see text] isomorphic copies of generalized Petersen graph.

scholarly journals Eternal Domination of Generalized Petersen Graph

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ramy Shaheen ◽  
Ali Kassem
Keyword(s):  
Infinite Series ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Adjacent Vertex ◽  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Dominating Set Problem

An eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks; an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the m -eternal domination model. The size of the smallest m -eternal dominating set is called the m -eternal domination number and is denoted by γ m ∞ G . In this paper, we find γ m ∞ P n , 1 and γ m ∞ P n , 3 for n ≡ 0   mod   4 . We also find upper bounds for γ m ∞ P n , 2 and γ m ∞ P n , 3 when n is arbitrary.

Perfect 2-colorings of the generalized Petersen graph

2016 ◽  
Vol 126 (3) ◽  
pp. 289-294 ◽  
Author(s):  
MEHDI ALAEIYAN ◽  
HAMED KARAMI
Keyword(s):  
Petersen Graph ◽  
Generalized Petersen Graph

scholarly journals Strong Edge Coloring of Generalized Petersen Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1265
Author(s):  
Ming Chen ◽  
Lianying Miao ◽  
Shan Zhou
Keyword(s):  
Edge Coloring ◽  
Petersen Graph ◽  
Chromatic Index ◽  
Induced Matching ◽  
Generalized Petersen Graph ◽  
Generalized Petersen Graphs ◽  
Strong Chromatic Index ◽  
Color Class ◽  
Proper Edge Coloring ◽  
Petersen Graphs

A strong edge coloring of a graph G is a proper edge coloring such that every color class is an induced matching. In 2018, Yang and Wu proposed a conjecture that every generalized Petersen graph P(n,k) with k≥4 and n>2k can be strong edge colored with (at most) seven colors. Although the generalized Petersen graph P(n,k) is a kind of special graph, the strong chromatic index of P(n,k) is still unknown. In this paper, we support the conjecture by showing that the strong chromatic index of every generalized Petersen graph P(n,k) with k≥4 and n>2k is at most 9.

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