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

Tatiana Dovzhenok

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 $3$.

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

Abstract

Let and be simple finite undirected graphs. A graph is called -irregular if any two of its distinct vertices belong to different numbers of copies of in . According to the strong conjecture about -irregular graphs (Dovzhenok, Filuta, Chuhai), for any connected graph of order , there exist infinitely many -irregular graphs. In the present paper, the strong conjecture about -irregular graphs is confirmed in the class of graphs of diameter . It is proved that for every graph of diameter , there exists an infinite series of -irregular graphs of diameter .
Paper Structure (8 sections, 18 theorems, 50 equations, 4 figures)

This paper contains 8 sections, 18 theorems, 50 equations, 4 figures.

Table of Contents

  1. Introduction and statement of the problem
  2. Basic graph constructions
  3. Graph $A_{2l-1}$
  4. Graph $F_{2l}$
  5. Proof of the Theorem \ref{['th4']}
  6. $\langle n,t \rangle$-condition
  7. Proof of the Theorem \ref{['th4']}
  8. Conclusion

Key Result

Theorem 1

Let $K_{n}$ and $K_{1,n-1}$ be a complete graph and a star of order $n \geqslant 3$. Then, there exist graphs that are $K_{n}$-irregular and $K_{1,n-1}$-irregular, respectively.

Figures (4)

  • Figure 1: Graph $A_{2l-1}$.
  • Figure 2: Graph $H=A_{2l-1}\backslash (i+1,l+i)$.
  • Figure 3: Graph $F_{2l}$.
  • Figure 4: Diagram illustrating the evolution of $F$-degrees of vertices in graph $A_{2l-1}$ when transitioning to graph $F_{2l}$.

Theorems & Definitions (47)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Conjecture 1
  • Theorem 2
  • Conjecture 2
  • Theorem 3
  • Theorem 4
  • Definition 3
  • Definition 4
  • ...and 37 more