{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,16]],"date-time":"2024-06-16T07:53:09Z","timestamp":1718524389934},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"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 $C$.<\/jats:p>","DOI":"10.37236\/507","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:56:57Z","timestamp":1578697017000},"source":"Crossref","is-referenced-by-count":9,"title":["On a Conjecture Concerning the Petersen Graph"],"prefix":"10.37236","volume":"18","author":[{"given":"Donald","family":"Nelson","sequence":"first","affiliation":[]},{"given":"Michael D.","family":"Plummer","sequence":"additional","affiliation":[]},{"given":"Neil","family":"Robertson","sequence":"additional","affiliation":[]},{"given":"Xiaoya","family":"Zha","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2011,1,19]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p20\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p20\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T18:17:48Z","timestamp":1579285068000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p20"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1,19]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/507","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1,19]]}}}