A note on Haynes-Hedetniemi-Slater Conjecture

TL;DR

This paper confirms the Haynes-Hedetniemi-Slater Conjecture for graphs with minimum degree at least 4, establishing an upper bound on the domination number relative to the graph's order.

Contribution

It proves the conjecture for all graphs with minimum degree , extending previous partial results and confirming the conjecture's validity in this case.

Findings

The conjecture holds for -degree graphs.

The domination number is bounded by (n) for these graphs.

The result generalizes previous partial proofs.

Abstract

We notice that Haynes-Hedetniemi-Slater Conjecture is true (i.e. $\gamma(G) \leq \frac{\delta}{3\delta -1}n$ for every graph $G$ of size $n$ with minimum degree $\delta \geq 4$, where $\gamma(G)$ is the domination number of $G$). Because the conjecture for $\delta =6$ follows from the estimate n (1 - \prod_{i= 1}^[\delta + 1} (\delta i)/(\delta i + 1) by W. E. Clark, B. Shekhtman, S. Suen [Upper bounds of the Domination Number of a Graph, Congressus Numerantium, 132 (1998), pp. 99-123.]