On the size of identifying codes in triangle-free graphs

TL;DR

This paper establishes an upper bound on the size of minimum identifying codes in connected triangle-free graphs with maximum degree at least 3, showing the bound is asymptotically tight and proposing conjectures for further improvements.

Contribution

It provides a new upper bound on the minimum size of identifying codes in triangle-free graphs, extending understanding of their structure and bounds.

Findings

Bound $ ext{M}(G) o n - rac{n}{ riangle + o( riangle)}$ for such graphs.

The bound is tight up to constants, matching known classes like $( riangle-1)$-ary trees.

Improved bounds are given for specific subclasses of triangle-free graphs.

Abstract

In an undirected graph $G$, a subset $C\subseteq V(G)$ such that $C$ is a dominating set of $G$, and each vertex in $V(G)$ 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 $\M(G)$ 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 $\Delta\geq 3$, $\M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}$. 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 $n-\tfrac{n}{\Delta-1+o(1)}$. We also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant $c$ such that the bound $\M(G)\le n-\tfrac{n}{\Delta}+c$ holds for any nontrivial connected identifiable graph $G$.