A permutation graph is a cubic graph admitting a 1-factor M whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if e is an edge of M such that every 4-cycle containing an edge of M contains e, then e is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-conn...

Let P be the Petersen graph. The main results of this paper are the discovery of infinite families of chromatically equivalent pairs of P homeomorphs and the discovery of infinite families of flow equivalent pairs of P amallamorphs. In particular, three families of P homeomorphs with 8 parameters, five families with 7 parameters and many families with fewer parameters are obtained. Also one fam...

By hypotheses (a) and (c), if {i, j} is a 2-element subset of the type set I of Γ, then there are flags F of co-type {i, j}, such that res(F ) is a rank 2 geometry over the type set {i, j}, and all such {i, j}-residues are isomorphic. Hence, such a geometry Γ can be described by a diagram, in which the types (or even the exact isomorphism types) of all rank-2-residues are listed; this is often ...

A (1, 2)-eulerian weight w of a cubic graph is hamiltonian if every faithful circuit cover of the graph with respect to w is a set of two Hamilton circuits. Let G be a 3-connected cubic graph containing no Petersen-minor. It is proved in this paper that G admits a Hamilton weight if and only if G can be obtained from K4 by a series of 4 ↔ Y -operations. As a byproduct of the proof of the main t...

Robertson conjectured that the only 3-connected, internally 4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph. We provide a counterexample to this conjecture.

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.

This article has restudied the Petersen family in Graph Theory. It discussed process of establishing this family. discussion leads to discovering some new properties family's members.