scholarly journals On Generalizations of the Petersen Graph and the Coxeter Graph

10.37236/3759 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Marko Orel
Keyword(s):  
Petersen Graph ◽  
Coxeter Graph

In this note we consider two related infinite families of graphs, which generalize the Petersen and the Coxeter graph. The main result proves that these graphs are cores. It is determined which of these graphs are vertex/edge/arc-transitive or distance-regular. Girths and odd girths are computed. A problem on hamiltonicity is posed.A Corrigendum for this paper was added on August 19, 2017.

Flip-Flops in Hypohamiltonian Graphs

1973 ◽  
Vol 16 (1) ◽  
pp. 33-41 ◽  
Author(s):  
V. Chvátal
Keyword(s):  
Infinite Sequence ◽  
Systematic Search ◽  
Petersen Graph ◽  
Hamiltonian Graph ◽  
Coxeter Graph

Throughout this note, we adopt the graph-theoretical terminology and notation of Harary [3]. A graph G is hypohamiltonianif G is not hamiltonian but the deletion of any point u from G results in a hamiltonian graph G-u. Gaudin, Herz, and Rossi [2] proved that the smallest hypohamiltonian graph is the Petersen graph. Using a computer for a systematic search, Herz, Duby, and Vigué [4] found that there is no hypohamiltonian graph with 11 or 12 points. However, they found one with 13 and one with 15 points. Sousselier [4] and Lindgren [5] constructed independently the same sequence of hypohamiltonian graphs with 6k+10 points. Moreover, Sousselier found a cubic hypohamiltonian graph with 18 points. This graph and the Petersen graph were the only examples of cubic hypohamiltonian graphs until Bondy [1] constructed an infinite sequence of cubic hypohamiltonian graphs with 12k+10 points. Bondy also proved that the Coxeter graph [6], which is cubic with 28 points, is hypohamiltonian.

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.

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