Fan-type Theorem for a Long Path Passing Through a Specified Vertex

Abstract

Let $G$ be a $2$-connected graph with maximum degree $\mathrm{\Delta }(G)\ge d$, and let $x$ and $z$ be distinct vertices of $G$. Let $W$ be a subset of $V(G)\setminus \{x,z\}$ such that $|W|\le d–1$. Hirohata proved that if $max\{d_G(u),d_G(v)\}\ge d$ for every pair of vertices $u,v\in V(G)\setminus \{x,z\}\cup W$ such that $d_G(u,v)=2$, then $x$ and $z$ are joined by a path of length at least $d–|W|$. In this paper, we show that if $G$ satisfies the conditions of Hirohata’s theorem, then for any given vertex $y$ such that $d_G(y)\ge d$, $x$ and $z$ are joined by a path of length at least $d–|W|$ which contains $y$.