Year: 2026
Type: preprint
Abstract: For a given graph $F$, a graph $G$ is $F$-irregular if all its vertices have pairwise distinct $F$-degrees, where the $F$-degree of a vertex $v$ is the number of subgraphs of $G$ containing $v$ and isomorphic to $F$. While the existence of $F$-irregular graphs has been actively studied for connected graphs $F$, the case where $F$ is disconnected remains entirely unexplored. In this paper, we provide a complete solution to this problem. We prove that for every disconnected graph $F$ with $|E(F)| \ge 2$, there exist infinitely many $F$-irregular graphs. Conversely, if $|E(F)| \le 1$, we show that no non-trivial $F$-irregular graph exists.
Source: arXiv (Cornell University)
Author: Dovzhenok, Tatiana
Institution: Francisk Skorina Gomel State University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221791778