The 1/3-conjectures for domination in cubic graphs

TL;DR

This paper advances the understanding of domination in cubic graphs by proving two conjectures related to the domination number being at most one-third of the vertices under specific girth and bipartite conditions.

Contribution

The paper proves Verstraete's conjecture for cubic graphs with girth at least 6 excluding 7- and 8-cycles, and Kostochka's conjecture for bipartite cubic graphs without 4- or 8-cycles.

Findings

Verstraete's conjecture holds for cubic graphs with girth ≥6 excluding 7- and 8-cycles.

Kostochka's conjecture holds for bipartite cubic graphs without 4- and 8-cycles.

Provides new bounds on domination numbers under specific girth and bipartite constraints.

Abstract

A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to a vertex in S . The domination number of G, denoted by $\gamma$(G), is the minimum cardinality of a dominating set in G. In a breakthrough paper in 2008, L{\"o}wenstein and Rautenbach proved that if G is a cubic graph of order n and girth at least 83, then $\gamma$(G) $\le$ n/3. A natural question is if this girth condition can be lowered. The question gave birth to two 1/3-conjectures for domination in cubic graphs. The first conjecture, posed by Verstraete in 2010, states that if G is a cubic graph on n vertices with girth at least 6, then $\gamma$(G) $\le$ n/3. The second conjecture, first posed as a question by Kostochka in 2009, states that if G is a cubic, bipartite graph of order n, then $\gamma$(G) $\le$n/3. In this paper, 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.