Journal of Graph Theory· 2026Q1
The 13‐Conjectures for Domination in Cubic Graphs
- 1citations
- Q1SCImago
- 2026year
Short summary
This paper proves Verstraete's conjecture (domination number of a cubic graph with girth >= 6 is at most n/3) for cubic graphs without 7- or 8-cycles, and Kostochka's conjecture (domination number of a cubic bipartite graph with order n is at most n/3) for bipartite graphs without 4- or 8-cycles.
AI-generated from the title and abstract; the full text is not read.
Key points
- Proves Verstraete's conjecture for cubic graphs without 7- or 8-cycles.
- Proves Kostochka's conjecture for cubic bipartite graphs without 4- or 8-cycles.
- Both conjectures relate the domination number of cubic graphs to n/3, where n is the number of vertices.
AI-generated from the title and abstract; the full text is not read.
Abstract
ABSTRACT A set of vertices in a graph is a dominating set of if every vertex not in is adjacent to a vertex in . The domination number of , denoted by , is the minimum cardinality among all dominating sets in . In a breakthrough paper in 2008, Löwenstein and Rautenbach proved that if is a cubic graph of order and girth at least 83, then . A natural question is if this girth condition can be lowered. The question gave birth to two ‐conjectures for domination in cubic graphs. The first conjecture, posed by Verstraete in 2010, states that if is a cubic graph on vertices with girth at least 6, then . The second conjecture, first posed as a question by Kostochka in 2009, states that if is a cubic, bipartite graph of order , then . In this article, we prove Verstraete's conjecture when there is no 7‐cycle and no 8‐cycle, and we prove the Kostochka's related conjecture for bipartite graphs when there is no 4‐cycle and no 8‐cycle.
The authors' abstract, as published at the source. Journal of Graph Theory, 2026 · DOI ↗
Continue with a free account
Ask the paper: 3 free questions a day about this paper; save it, get its citation, new summaries every day for your field. Takeaways are Premium.
Continue free on the webSign in with Google or Apple; no card needed. You come back to this paper.
On your phone:
Field: Computational Theory and Mathematics
Computational Theory and MathematicsComputer Science