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EnglishA snark is a cubic cyclically 4-edge connected graph with edge chromatic number four and girth at least five. We say that a graph G is odd 2-factored if for each 2-factor F of G each cycle of F is odd. Some of the authors conjectured in Abreu et al. (2012) [4] that a snark G is odd 2-factored if and only if G is the Petersen graph, Blanuša 2, or a flower snark J(t), with t≥5 and odd. Brinkmann et al. (2013) [10] have obtained two counterexamples that disprove this conjecture by performing an exhaustive computer search of all snarks of order n≤36.In this paper, we present a method for constructing odd 2-factored snarks. In particular, we independently construct the two odd 2-factored snarks that yield counterexamples to the above conjecture. Moreover, we approach the problem of characterizing odd 2-factored snarks furnishing a partial characterization of cyclically 4-edge connected odd 2-factored snarks. Finally, we pose a new conjecture regarding odd 2-factored snarks.
Odd 2-factored snarks / Abreu, Marien; Labbate, Domenico; Rizzi, Romeo; Sheehan, John. - In: EUROPEAN JOURNAL OF COMBINATORICS. - ISSN 0195-6698. - STAMPA. - 36(2014), pp. 460-472.
| Titolo: | Odd 2-factored snarks |
| Autori: | |
| Data di pubblicazione: | 2014 |
| Rivista: | |
| Citazione: | Odd 2-factored snarks / Abreu, Marien; Labbate, Domenico; Rizzi, Romeo; Sheehan, John. - In: EUROPEAN JOURNAL OF COMBINATORICS. - ISSN 0195-6698. - STAMPA. - 36(2014), pp. 460-472. |
| Abstract: | A snark is a cubic cyclically 4-edge connected graph with edge chromatic number four and girth at least five. We say that a graph G is odd 2-factored if for each 2-factor F of G each cycle of F is odd. Some of the authors conjectured in Abreu et al. (2012) [4] that a snark G is odd 2-factored if and only if G is the Petersen graph, Blanuša 2, or a flower snark J(t), with t≥5 and odd. Brinkmann et al. (2013) [10] have obtained two counterexamples that disprove this conjecture by performing an exhaustive computer search of all snarks of order n≤36.In this paper, we present a method for constructing odd 2-factored snarks. In particular, we independently construct the two odd 2-factored snarks that yield counterexamples to the above conjecture. Moreover, we approach the problem of characterizing odd 2-factored snarks furnishing a partial characterization of cyclically 4-edge connected odd 2-factored snarks. Finally, we pose a new conjecture regarding odd 2-factored snarks. |
| Handle: | http://hdl.handle.net/11562/933308 |
| Appare nelle tipologie: | 01.01 Articolo in Rivista |
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