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

Tatiana Dovzhenok, Ilya Lukashenko, Yahor Filiuta

TL;DR

The paper advances irregularity theory for oriented graphs by proving that for every $n\ge3$, there exist infinite families of $\overrightarrow{C_n}$-irregular graphs. It develops two main construction frameworks: an $A_{2l+2,n}$-based scheme yielding irregularity for all $n\ge5$, and explicit families of $\overrightarrow{B_k}$ and $\overrightarrow{D_k}$ graphs that cover $\overrightarrow{C_4}$- and $\overrightarrow{C_3}$-irregular cases, respectively. It establishes sharp minimal-order results: no nontrivial $\overrightarrow{C_4}$-irregular graphs exist for $k<7$, and no nontrivial $\overrightarrow{C_3}$-irregular graphs exist for $k<10$, with constructive extensions to all larger orders. The work culminates in a main theorem confirming the conjecture that infinitely many $\overrightarrow{C_n}$-irregular oriented graphs exist for every $n\ge3$, thus extending irregularity concepts to broader directed-graph settings.

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 circle 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 nothing else. We also construct $\overrightarrow{C_4}$-irregular graphs of any order starting from $7$ and prove that there is no non-trivial $\overrightarrow{C_4}$-irregular graph of order less than $7$.

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

TL;DR

The paper advances irregularity theory for oriented graphs by proving that for every , there exist infinite families of -irregular graphs. It develops two main construction frameworks: an -based scheme yielding irregularity for all , and explicit families of and graphs that cover - and -irregular cases, respectively. It establishes sharp minimal-order results: no nontrivial -irregular graphs exist for , and no nontrivial -irregular graphs exist for , with constructive extensions to all larger orders. The work culminates in a main theorem confirming the conjecture that infinitely many -irregular oriented graphs exist for every , thus extending irregularity concepts to broader directed-graph settings.

Abstract

Let and be simple finite oriented graphs (without symmetric arcs). A graph is called -irregular if any two distinct vertices in belong to a different number of subgraphs of isomorphic to . In this paper, we investigate the problem of the existence of -irregular graphs, where is an oriented circle of order (a strongly connected oriented graph that is formed from a simple undirected cycle on vertices by orienting each of its edges). For every integer , we prove that there exists an infinite family of -irregular graphs. In addition, we show that the order of a non-trivial -irregular graph can be any integer not less than and nothing else. We also construct -irregular graphs of any order starting from and prove that there is no non-trivial -irregular graph of order less than .

Paper Structure

This paper contains 15 sections, 25 theorems, 39 equations, 11 figures, 5 tables.

Table of Contents

  1. Introduction
  2. $\overrightarrow{C _n }$-irregular graphs in case $n \ge 5$
  3. On the order of $\overrightarrow{C _4 }$-irregular graphs
  4. $\overrightarrow{C_4}$-irregular graph of order 7
  5. $\overrightarrow{C_4}$-irregular graph of order 8
  6. $\overrightarrow{C _4 }$-irregular graph of order 10
  7. $\overrightarrow{C_4}$-irregular graphs of even order greater than 10
  8. Criterion for the existence of a $\overrightarrow{C _4 }$-irregular graph of order $k$
  9. On $\overrightarrow{C _3 }$-irregular graphs
  10. $\overrightarrow{C _3 }$-irregular graph of order 10
  11. $\overrightarrow{C _3 }$-irregular graph of order 12
  12. $\overrightarrow{C _3 }$-irregular graphs of order $k \ge 14$, $k \equiv 2 \, (mod \ 4)$
  13. $\overrightarrow{C _3 }$-irregular graphs of order $k \ge 16$, $k \equiv 0 \, (mod \ 4)$
  14. Criterion for the existence of a $\overrightarrow{C _3 }$-irregular graph of order $k$
  15. Main result

Key Result

Lemma 1

Let $l, n$ be integers such that $l \ge n \ge 5$. Then the $\overrightarrow{C_n}$-degrees of the vertices in the graph $\overrightarrow{A_{2l+2,\, n}}$ are equal to $1) \ a_i=iC_{l-2}^{n-4} \ \ \forall i \in \{1,2,...,l-1\};$$2) \ a_l=lC_{l-1}^{n-3};$$3) \ a_i=(2l-i)C_{l-2}^{n-4}+C_{l-1}^{n-3}+1 \

Figures (11)

  • Figure 1: $K_3$-irregular graph $D_8$.
  • Figure 2: Graph $\overrightarrow{A_{2l+2,\, n}}$.
  • Figure 3: Graph $\overrightarrow{B_7}$.
  • Figure 4: Graph $\overrightarrow{B_8}$.
  • Figure 5: Graph $\overrightarrow{B_{10}}$.
  • ...and 6 more figures

Theorems & Definitions (68)

  • Definition 1
  • Conjecture 1
  • Conjecture 2: Strong conjecture about $F$-irregular graphs
  • Conjecture 3: Strong conjecture about $F$-irregular oriented graphs
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • proof
  • Proposition 1
  • ...and 58 more