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{\bf Donald Nelson, Michael D. Plummer, Neil Robertson and Xiaoya Zha}
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{\bf On a Conjecture Concerning the Petersen Graph}
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Robertson has 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 prove this
conjecture in the special case where the graphs involved are also
cubic. Moreover, this proof does not require the
internal-4-connectivity assumption. An example is then presented to
show that the assumption of internal 4-connectivity cannot be
dropped as an hypothesis in the original conjecture.
We then summarize our results aimed toward the solution of the
conjecture in its original form. In particular, let $G$ be any
3-connected internally-4-connected graph of girth 5 in which every odd
cycle of length greater than 5 has a chord. If $C$ is any girth cycle
in $G$ then $N(C)\backslash V(C)$ cannot be edgeless, and if $N(C)
\backslash V(C)$ contains a path of length at least 2, then the
conjecture is true. Consequently, if the conjecture is false and $H$
is a counterexample, then for any girth cycle $C$ in $H$, $N(C)
\backslash V(C)$ induces a nontrivial matching $M$ together with an
independent set of vertices. Moreover, $M$ can be partitioned into
(at most) two disjoint non-empty sets where we can precisely describe
how these sets are attached to cycle $C$.
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