Path-factors involving paths of order seven and nine

TL;DR

This paper establishes new sufficient conditions based on component counts after vertex removal for the existence of path-factors involving paths of lengths seven and nine in graphs.

Contribution

It introduces two novel theorems providing criteria for graphs to contain specific path-factors involving paths of order seven and nine.

Findings

Conditions guarantee the existence of P2 and P7-factors.

Conditions guarantee the existence of P2 and P9-factors.

Provides a new approach to path-factor characterization.

Abstract

In this paper, we show the following two theorems (here $c_{i}(G-X)$ is the number of components $C$ of $G-X$ with $|V(C)|=i$): (i)~If a graph $G$ satisfies $c_{1}(G-X)+\frac{1}{3}c_{3}(G-X)+\frac{1}{3}c_{5}(G-X)\leq \frac{2}{3}|X|$ for all $X\subseteq V(G)$, then $G$ has a $\{P_{2},P_{7}\}$-factor. (ii)~If a graph $G$ satisfies $c_{1}(G-X)+c_{3}(G-X)+\frac{2}{3}c_{5}(G-X)+\frac{1}{3}c_{7}(G-X)\leq \frac{2}{3}|X|$ for all $X\subseteq V(G)$, then $G$ has a $\{P_{2},P_{9}\}$-factor.