| Title: | Girth, oddness, and colouring defect of snarks |
| Other Titles: | Obvod, lichost, a barevný defekt snarků |
| Authors: | Karabáš, Ján Máčajová, Edita Nedela, Roman Škoviera, Martin |
| Citation: | KARABÁŠ, J. MÁČAJOVÁ, E. NEDELA, R. ŠKOVIERA, M. Girth, oddness, and colouring defect of snarks. DISCRETE MATHEMATICS, 2022, roč. 345, č. 11, s. nestránkováno. ISSN: 0012-365X |
| Issue Date: | 2022 |
| Publisher: | Elsevier |
| Document type: | článek article |
| URI: | 2-s2.0-85132327227 http://hdl.handle.net/11025/49639 |
| ISSN: | 0012-365X |
| Keywords: | snark, lichost, obvod, barevný defekt |
| Keywords in different language: | snark, oddness, girth, colouring defect |
| Abstract: | Barevný defekt kubického grafu je mnimální počet hran nepokrytý třema perfektními párováními. Kubický graf má defekt 0 právě tehdy, když je hranově 3-obarvitelný. Naším cílem je studovat souvislost obvodu a lichosti kubického grafu a barevného defektu. Skonstruujeme nekonečně mnoho 5-cyklicky souvislých kubických grafů lichosti 2 libovolně velkého obvodu a libovolne velkého barevného defektu |
| Abstract in different language: | The colouring defect of a cubic graph, introduced by Steffen in 2015, is the minimum number of edges that are left uncovered by any set of three perfect matchings. Since a cubic graph has defect 0 if and only if it is 3-edge-colourable, this invariant can measure how much a cubic graph differs from a 3-edge-colourable graph. Our aim is to examine the relationship of colouring defect to oddness, an extensively studied measure of uncolourability of cubic graphs, defined as the smallest number of odd circuits in a 2factor. We show that there exist cyclically 5-edge-connected snarks (cubic graphs with no 3-edge-colouring) of oddness 2 and arbitrarily large colouring defect. This result is achieved by means of a construction of cyclically 5-edge-connected snarks with oddness 2 and arbitrarily large girth. The fact that our graphs are cyclically 5-edge-connected significantly strengthens a similar result of Jin and Steffen (2017), which only guarantees graphs with cyclic connectivity at most 3. At the same time, our result improves Kochol's original construction of snarks with large girth (1996) in that it provides infinitely many nontrivial snarks of any prescribed girth g >= 5, not just girth at least g. |
| Rights: | Plný text je přístupný v rámci univerzity přihlášeným uživatelům. © Elsevier |
| Appears in Collections: | Články / Articles (NTIS) OBD |
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