Graphs with $\{P_3,P_4,P_5\}$-factors in terms of size and spectral radius

TL;DR

This paper establishes size and spectral radius conditions that guarantee the existence of a spanning subgraph composed of paths of lengths 3, 4, or 5 in connected graphs.

Contribution

It provides new sufficient conditions based on size and spectral radius for the existence of specific path-factor structures in graphs.

Findings

Size condition ensures a -factor in graphs with minimum degree .

Spectral radius condition guarantees a -factor in graphs with minimum degree .

Abstract

Let $G$ be a connected graph of order $n$. A $\{P_3,P_4,P_5\}$-factor is a spanning subgraph $H$ of $G$ such that every component of $H$ is isomorphic to an element of $\{P_3,P_4,P_5\}$. In this paper, we establish a sufficient condition on the size of the graph $G$ with minimum degree $\delta$ to have a $\{P_3, P_4, P_5\}$-factor. Subsequently, we provide another sufficient condition on the adjacency spectral radius, ensuring that a connected graph $G$ with minimum degree $\delta$ contains a $\{P_3, P_4, P_5\}$-factor.