Proof of the strong conjecture about $F$-irregular graphs in the class of graphs $\{F\}$ of diameter $2$

TL;DR

This paper confirms the strong conjecture that infinitely many $F$-irregular graphs exist for any connected graph $F$ of diameter 2, specifically within the class of graphs of diameter 2, by constructing an infinite series of such graphs.

Contribution

It proves the strong conjecture for the class of graphs with diameter 2 by demonstrating an infinite series of $F$-irregular graphs of diameter 3.

Findings

Confirmed the strong conjecture for diameter 2 graphs.

Constructed an infinite series of $F$-irregular graphs of diameter 3.

Extended understanding of $F$-irregular graphs in diameter-restricted classes.

Abstract

Let $F$ and $G$ be simple finite undirected graphs. A graph $G$ is called $F$-irregular if any two of its distinct vertices belong to different numbers of copies of $F$ in $G$. According to the strong conjecture about $F$-irregular graphs (Dovzhenok, Filuta, Chuhai), for any connected graph $F$ of order $|F|\geqslant 3$, there exist infinitely many $F$-irregular graphs. In the present paper, the strong conjecture about $F$-irregular graphs is confirmed in the class of graphs $\{F\}$ of diameter $2$. It is proved that for every graph $F$ of diameter $2$, there exists an infinite series of $F$-irregular graphs of diameter $3$.