A proof of the $\frac{3}{8}$-conjecture for independent domination in cubic graphs

TL;DR

The paper proves that every connected cubic graph $G$ with $G \not\cong K_{3,3}$ and $G \not\cong C_5 \Box K_2$ satisfies $i(G) \le \frac{3}{8}n$, resolving a longstanding conjecture. The authors introduce an $\,\Omega$-weight framework combining a vertex-weight function with a structural weight that accounts for problematic substructures (bad graphs and troublesome configurations). They establish the general bound $8i(G) \le \Omega(G)$ for subcubic graphs with no $K_{3,3}$-components and no $(C_5 \Box K_2)$-components, and then specialize to cubic graphs where the structural weight vanishes, yielding the desired $i(G) \le \frac{3}{8}n$. The argument hinges on careful structural analysis, weight accounting, and elimination of potential counterexamples through four-part casework, culminating in the proof of the conjecture and highlighting the tightness via explicit extremal graph families.

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~$S$. An independent dominating set in $G$ is a dominating set of $G$ with the additional property that it is an independent set. The domination number, $γ(G)$, and the independent domination number, $i(G)$, are the minimum cardinalities among all dominating sets and independent dominating sets in $G$, respectively. By definition, $γ(G) \le i(G)$ for all graphs $G$. Let $G$ be a connected cubic graph of order~$n$. In 1996 Reed [Combin.\ Probab.\ Comput.\ 5 (1996), 277--295] proved a breakthrough result that $γ(G) \le \frac{3}{8}n$. We prove the stronger result that if $G$ is different from $K_{3,3}$ and the $5$-prism $C_5 \, \Box \, K_2$, then $i(G) \le \frac{3}{8}n$. This proves a known conjecture. The bound is tight in the sense that there are infinite families of connected cubic graphs that achieve equality in this bound.

Figures (56)

• Figure 1: The graphs $K_{2,3}$ and $C_5 \, \Box \, K_2$.
• Figure 2: Two infinite families of cubic graphs $G$ of order $n$ satisfying $i(G) = \frac{3}{8}n$
• Figure 3: An infinite family of cubic graphs $G$ of order $n$ satisfying $i(G) = \frac{3}{8}n$
• Figure 4: The base graph $B_1$
• Figure 5: The operation ${\cal O}_1$

Theorems & Definitions (71)

• Theorem 1
• Theorem 2
• Conjecture 1
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 1
• Proposition 1
• Proposition 2