On the size of identifying codes in triangle-free graphs

Abstract

In an undirected graph $G$, a subset $C\subseteq V\left(G\right)$ such that $C$ is a dominating set of $G$, and each vertex in $V\left(G\right)$ is dominated by a distinct subset of vertices from $C$, is called an identifying code of $G$. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph $G$, let ${\gamma }^{\mathrm{ID}}\left(G\right)$ be the minimum cardinality of an identifying code in $G$. In this paper, we show that for any connected identifiable triangle-free graph $G$ on $n$ vertices having maximum degree $\Delta \ge 3$, ${\gamma }^{\mathrm{ID}}\left(G\right)\le n-\frac{n}{\Delta +o\left(\Delta \right)}$. This bound is asymptotically tight up to constants due to various classes of graphs including $(\Delta -1)$-ary trees, which are known to have their minimum identifying code of size $n-\frac{n}{\Delta -1+o\left(1\right)}$. We also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant $c$ such that the bound ${\gamma }^{\mathrm{ID}}\left(G\right)\le n-\frac{n}{\Delta }+c$ holds for any nontrivial connected identifiable graph $G$.