Issue |
RAIRO-Oper. Res.
Volume 57, Number 5, September-October 2023
|
|
---|---|---|
Page(s) | 2521 - 2526 | |
DOI | https://doi.org/10.1051/ro/2023143 | |
Published online | 06 October 2023 |
Reducible 3-critical graphs
University of Cape Town, Cape Town, South Africa
* Corresponding author: imran.allie@uct.ac.za
Received:
31
March
2023
Accepted:
7
September
2023
In this paper, we provide further insights into the reducibility of 3-critical graphs. A graph is 3-critical if it is class two, that is, has chromatic index 4, and the removal of any one edge renders a graph with chromatic index 3. We consider two types of reductions: the suppression of a 2-connected subgraph into a single edge; and the suppression of a 3-connected subgraph into a single vertex. That is, in cases where the 2- or 3-connected subgraph is cubic or strictly subcubic. We show that every cyclically 2-connected 3-critical graph can be reduced to a smaller 3-critical graph using this method. We also prove that every 3-critical graph which is cyclically k-connected with k = 2 or k = 3, contains a 3-critical minor which is cyclically k-connected with k ≥ 4.
Mathematics Subject Classification: 05C15 / 05C83
Key words: k-critical graphs / cubic graphs / snarks / edge-colourings
© The authors. Published by EDP Sciences, ROADEF, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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