Perfect matchings in highly cyclically connected regular graphs

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23520%2F22%3A43964193" target="_blank" >RIV/49777513:23520/22:43964193 - isvavai.cz</a>
Result on the web
<a href="https://onlinelibrary.wiley.com/doi/full/10.1002/jgt.22764" target="_blank" >https://onlinelibrary.wiley.com/doi/full/10.1002/jgt.22764</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1002/jgt.22764" target="_blank" >10.1002/jgt.22764</a>

Result language
angličtina
Original language name
Perfect matchings in highly cyclically connected regular graphs
Original language description
A leaf matching operation on a graph consists of removing a vertex of degree 1 together with its neighbour from the graph. Let G be a d‐regular cyclically (d k − 1+2 )‐ edge‐connected graph of even order, where k ≥ 0 and d ≥ 3. We prove that for any given set X of d k − 1 + edges, there is no 1‐factor of G avoiding X if and only if either an isolated vertex can be obtained by a series of leaf matching operations in G − X, or G − X has an independent set that contains more than half of the vertices of G. To demonstrate how to check the conditions of the theorem we prove several statements on 2‐factors of cubic graphs. For k ≥ 3, we prove that given a cyclically (4k − 5)‐edge‐connected cubic graphG and three paths of length k such that the distance between any two of them is at least 8k − 16, there is a 2‐factor of G that contains one of the paths. We provide a similar statement for two paths when k = 3 and k = 4. As a corollary we show that given a vertex v in a cyclically 7‐edge‐connected cubic graph, there is a 2‐factor such that v is in a circuit of length greater than 7.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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