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6-decomposition of snarks
Title 6-decomposition of snarks Author info Ján Karabáš, Edita Máčajová, Roman Nedela Author Karabáš Ján 1977- (34%) UMBFP12 - Inštitút matematiky a informatiky
Co-authors Máčajová Edita (33%)
Nedela Roman 1960- (33%) UMBFP10 - Katedra matematiky
Action International Workshop on Combinatorial Algorithms (IWOCA) . medzinárodný workshop , 20. , Hradec nad Moravicí , 28.06.-02.07.2009 Source document European Journal of Combinatorics. Vol. 34, no. 1 (2013), pp. 111-122. - London : Academic Press, 2013 Keywords matematika - mathematics snark 6-decomposition Language English systematics 51 Annotation A snark is a cubic graph with no proper $3$-edge-colouring. In 1996, Nedela and /v Skoviera proved the following theorem: Let G be a snark with an k-edge-cut, k>= 2, whose removal leaves two 3-edge-colourable components M and N. Then both M and N can be completed to two snarks $/tilde M$ and $/tilde N$ of order not exceeding that of G by adding at most $/kappa(k)$ vertices, where the number $/kappa(k)$ only depends on $k$. The known values of the function $/kappa(k)$ are $/kappa(2)=0$, $/kappa(3)=1$, $/kappa(4)=2$ (Goldberg, 1981), and $/kappa(5)=5$ (Cameron, Chetwynd, Watkins, 1987). The value $/kappa(6)$ is not known and is apparently difficult to calculate. In 1979, Jaeger conjectured that there are no 7-cyclically-connected snarks. If this conjecture holds true, then $/kappa(6)$ is the last important value to determine. The paper is aimed attacking the problem of determining $/kappa(6)$ by investigating the structure and colour properties of potential complements in $6$-decompositions of snarks. We find a set of $14$ complements that suffice to perform $6$-decompositions of snarks with at most $30$ vertices. We show that if this set is not complete to perform $6$-decompositions of all snarks, then $/kappa(6)/geq 20$ and there are strong restrictions on the structure of (possibly) missing complements. Public work category AFC No. of Archival Copy 25350 Repercussion category FIOL, M.A. - MAZZUOCCOLO, G. - STEFFEN, E. On measures of edge-uncolorability of cubic graphs : a brief survey and some new results. In arXiv:1702.07156 [math.CO] [online]. New York : Cornell Univerisity Library, 2017, [1-39][cit. 2018-22-02]. Dostupné na: https://arxiv.org/pdf/1702.07156.pdf
FIOL, M. A. - VILALTELLA, J. Some results on the structure of multipoles in the study of snarks. In Electronic journal of combinatorics. ISSN 1077-8926, 2015, vol. 22, no. 1.
FIOL, M. A. - MAZZUOCCOLO, G. - STEFFEN, E. Measures of edge-uncolorability of cubic graphs. In Electronic journal of combinatorics. ISSN 1077-8926, 2018, vol. 25, no. 4.
OZEKI, Kenta. Kempe equivalence classes of cubic graphs embedded on the projective plane. In Combinatorica. ISSN 0209-9683, 2022, vol. 42, no. 2, pp. 1451-1480.
MAZAK, Jan - RAJNIK, Jozef - SKOVIERA, Martin. Morphology of small snarks. In Electronic journal of combinatorics. ISSN 1077-8926, 2022, vol. 29, no. 4, pp. 1-46.
SKRINAROVA, Jarmila - DUDAS, Adam. Optimization of the functional decomposition of parallel and distributed computations in graph coloring with the use of high-performance computing. In IEEE access. ISSN 2169-3536, 2022, vol. 10, pp. 34996-35011.
DUDÁŠ, Adam - ŠKRINÁROVÁ, Jarmila - KISS, Adam. On graph coloring analysis through visualization. In Information and digital technologies 2021 : proceedings of the international conference, Žilina, 22.-24.06.2021. 1. vyd. Danvers : Institute of electrical and electronics engineers, 2021. ISBN 978-1-6654-3692-2, pp. 71-78.
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