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The 1/3-conjectures for domination in cubic graphs

Paul Dorbec, Michael Antony Henning

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 $γ$(G), is the minimum cardinality of a dominating set in G. In a breakthrough paper in 2008, L{ö}wenstein and Rautenbach proved that if G is a cubic graph of order n and girth at least 83, then $γ$(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 $γ$(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 $γ$(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.

The 1/3-conjectures for domination in cubic graphs

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 (G), is the minimum cardinality of a dominating set in G. In a breakthrough paper in 2008, L{ö}wenstein and Rautenbach proved that if G is a cubic graph of order n and girth at least 83, then (G) 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 (G) 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 (G) 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.
Paper Structure (29 sections, 8 theorems, 61 equations, 45 figures, 1 table)

This paper contains 29 sections, 8 theorems, 61 equations, 45 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Notation
  4. Statement of the main theorem
  5. Proof of the main theorem
  6. Fundamental structural properties
  7. The associated colored multigraph
  8. The green edges form a Matching, as do the red edges
  9. There are no long edges
  10. Key structural claims
  11. Maximal green-black alternating paths
  12. Structural properties of extremities of green-black alternating paths
  13. Structural properties of internal vertices of black-green alternating paths
  14. Red islands.
  15. Green islands.
  16. ...and 14 more sections

Key Result

Theorem 1

(KoSt09) If $G$ is a connected, cubic graph of order $n$, then $\gamma(G) \le \frac{5}{14}n = \left( \frac{1}{3} + \frac{1}{42} \right)n$.

Figures (45)

  • Figure 1: The generalized Petersen graph $P(7,2)$.
  • Figure 2: A colored multigraph in the proof of Lemma \ref{['c:no-green-red-green']}
  • Figure 3: A colored multigraph in the proof of Lemma \ref{['c:no-green-red-red-path']}
  • Figure 4: Colored multigraphs in the proof of Lemma \ref{['c:no-double-star-w-2-red']}
  • Figure 5: The two cases of Claim \ref{['c:claim23']}
  • ...and 40 more figures

Theorems & Definitions (137)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Conjecture 5
  • Conjecture 6
  • Theorem 7
  • Theorem 8
  • Corollary 9
  • Theorem 10
  • ...and 127 more