On prescribed Hamilton laceability of hybrid-faulty star graphs

Abstract

In this paper, we investigate Hamilton paths passing through a prescribed linear forest in a hybrid-faulty star graph. Let $M$ be a matching with $m$ edges of the star graph $S_n$, $F$ be an $f$-subset of $E(S_n-V\left(M\right))$, $\{x,y\}$ be a 2-subset of $V(S_n-V\left(M\right))$, and $L$ be a linear forest of $S_n-V\left(M\right)-F$. Suppose that $m+f\le n-4$, neither $x$ nor $y$ is an inner vertex of $L$, and $x,y$ are not located on the same component of $L$. We prove that, for any $w\in V(S_n-V\left(M\right))\setminus V\left(L\right)$, there exists a Hamilton path of $S_n-V\left(M\right)-F-w$ between $x$ and $y$ passing through $L$ provided that $x,y$ belong to the partite set not containing $w$, and $\left|E\left(L\right)\right|\le n-4-m-f$. As a consequence, if $x,y$ are from opposite partite sets and $\left|E\left(L\right)\right|\le n-4-m-f$ then $S_n-V\left(M\right)-F$ has a Hamilton path between $x$ and $y$ passing through $L$. We also proved that $S_n-V\left(M\right)-F$ has a Hamilton cycle passing through $L$ if $m+f\le n-3$ and $\left|E\left(L\right)\right|\le n-3-m-f$.