Some sufficient conditions for graphs to have component factors

TL;DR

This paper establishes sufficient conditions for graphs to contain specific component factors composed of stars and a special tree, and demonstrates the optimality of these conditions through extremal graph constructions.

Contribution

The paper introduces new sufficient conditions for the existence of certain component factors in graphs, extending previous factor theory results.

Findings

Derived conditions guarantee the existence of component factors in graphs.

Constructed extremal graphs to prove the sharpness of the main results.

Extended factor theory to include special tree components.

Abstract

Let $G$ denote a graph and $k\geq2$ be an integer. A $\{K_{1,1},K_{1,2},\ldots,K_{1,k},\mathcal{T}(2k+1)\}$-factor of $G$ is a spanning subgraph, whose every connected component is isomorphic to an element of $\{K_{1,1},K_{1,2},\ldots,K_{1,k},\mathcal{T}(2k+1)\}$, where $\mathcal{T}(2k+1)$ is one special family of tree. In this paper, we put forward some sufficient conditions for the existence of $\{K_{1,1},K_{1,2},\ldots,K_{1,k},\mathcal{T}(2k+1)\}$-factors in graphs. Furthermore, we construct some extremal graphs to show that the main results in this paper are best possible.