On $\overrightarrow{C_{n}}$-irregular oriented graphs

TL;DR

This paper studies the existence and properties of $ ightarrow C_n$-irregular oriented graphs, proving their infinite existence for all $n \,\geq\, 3$ and characterizing their orders for specific cases.

Contribution

It establishes the existence of infinite families of $ ightarrow C_n$-irregular graphs for all $n \ge 3$ and characterizes the possible orders for $ ightarrow C_3$ and $ ightarrow C_4$-irregular graphs.

Findings

Infinite families of $ ightarrow C_n$-irregular graphs exist for all $n \ge 3$.

Non-trivial $ ightarrow C_3$-irregular graphs can have any order ≥ 10.

$ ightarrow C_4$-irregular graphs exist for all orders ≥ 7, with none below that.

Abstract

Let $F$ and $G$ be simple finite oriented graphs (without symmetric arcs). A graph $G$ is called $F$-irregular if any two distinct vertices in $G$ belong to a different number of subgraphs of $G$ isomorphic to $F$. In this paper, we investigate the problem of the existence of $\overrightarrow{C_n}$-irregular graphs, where $\overrightarrow{C_n}$ is an oriented cycle of order $n$ (a strongly connected oriented graph that is formed from a simple undirected cycle $C_n$ on $n$ vertices by orienting each of its edges). For every integer $n \ge 3$, we prove that there exists an infinite family of $\overrightarrow{C_n}$-irregular graphs. In addition, we show that the order of a non-trivial $\overrightarrow{C_3}$-irregular graph can be any integer not less than $10$ and no others. We also construct $\overrightarrow{C_4}$-irregular graphs of any order at least $7$ and prove that there are no non-trivial $\overrightarrow{C_4}$-irregular graphs of order less than $7$.