Domination in 4-Regular Graphs With No Induced 4-Cycles

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 $\gamma \left(G\right)$, is the minimum cardinality of a dominating set in $G$. The $\frac{1}{3}$-conjecture for domination in 4-regular graphs states that if $G$ is a 4-regular graph of order $n$, then $\gamma \left(G\right)\le \frac{1}{3}n$. We prove this conjecture when $G$ has no induced 4-cycle.