Path-factors involving paths of order seven and nine

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 $c_1(G-X)+\frac{1}{3}{c}_3(G-X)+\frac{1}{3}{c}_5(G-X)\le \frac{2}{3}|X|$ for all $X\subseteq V(G)$, then $G$ has a $\{P_2,P_7\}$-factor. (ii)~If a graph $G$ satisfies $c_1(G-X)+c_3(G-X)+\frac{2}{3}{c}_5(G-X)+\frac{1}{3}{c}_7(G-X)\le \frac{2}{3}|X|$ for all $X\subseteq V(G)$, then $G$ has a $\{P_2,P_9\}$-factor.

arXiv comment29 pages, 4 figures

References in corpus (1)

• The existence of a path-factor without small odd paths