On a Conjecture Concerning the Petersen Graph

Abstract

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 $C$.

On a Conjecture Concerning the Petersen Graph: Part II

“…As also shown in [2], Robertson's Conjecture is true for many graphs; for example, for all cubic graphs and for those graphs which contain a girth cycle C such that N (C)−V (C) contains a path of length 2. (Here N (C) denotes the neighborhood of C, namely, the set of all vertices adjacent to at least one vertex of C.) It was also proved in [2] that given any 5-cycle C in a counterexample to Robertson's Conjecture, N (C) − V (C) cannot be an independent set.…”

“…A 3-connected graph G is said to be internally 4-connected if every cutset of three vertices is the neighborhood set of a fourth vertex. In [2] it was shown, by providing a counterexample, that the conclusion of the Robertson conjecture [3] stated in the Abstract is false, if the assumption that the graph is internally 4-connected is dropped.…”

A Note on Chromatic Number and Induced Odd Cycles

“…Robertson conjectured (see [16]) that the only 3-connected, internally 4-connected graph in G is the Petersen graph. This conjecture is true if the requirement on internally 4connectivity is replaced by cubic [16].…”

“…Robertson conjectured (see [16]) that the only 3-connected, internally 4-connected graph in G is the Petersen graph. This conjecture is true if the requirement on internally 4connectivity is replaced by cubic [16]. Plummer and Zha [17] presented a counterexample to Robertson's conjecture, and posed a few new questions including (1) whether every such graph has bounded chromatic number?…”

Proof of a conjecture of Plummer and Zha

“…(All graphs in this paper are finite, and have no loops or parallel edges.) Such graphs seem to be richly structured; indeed N. Robertson [3] proposed the conjecture that the Petersen graph is the only pentagraph that is three-connected and internally 4-connected, although this was disproved by M. Plummer and X. Zha [2] (see also [1]).…”