# Article

Keywords:
traceable graph; Hamiltonian graph; Hamiltonian-connected graph
Summary:
For a connected graph $G$ of order $n \ge 3$ and an ordering $s\: v_1$, $v_2, \cdots , v_n$ of the vertices of $G$, $d(s) = \sum _{i=1}^{n-1} d(v_i, v_{i+1})$, where $d(v_i, v_{i+1})$ is the distance between $v_i$ and $v_{i+1}$. The traceable number $t(G)$ of $G$ is defined by $t(G) = \min \left\rbrace d(s)\right\lbrace ,$ where the minimum is taken over all sequences $s$ of the elements of $V(G)$. It is shown that if $G$ is a nontrivial connected graph of order $n$ such that $l$ is the length of a longest path in $G$ and $p$ is the maximum size of a spanning linear forest in $G$, then $2n-2 - p \le t(G) \le 2n-2 - l$ and both these bounds are sharp. We establish a formula for the traceable number of every tree in terms of its order and diameter. It is shown that if $G$ is a connected graph of order $n \ge 3$, then $t(G)\le 2n-4$. We present characterizations of connected graphs of order $n$ having traceable number $2n-4$ or $2n-5$. The relationship between the traceable number and the Hamiltonian number (the minimum length of a closed spanning walk) of a connected graph is studied. The traceable number $t(v)$ of a vertex $v$ in a connected graph $G$ is defined by $t(v) = \min \lbrace d(s)\rbrace$, where the minimum is taken over all linear orderings $s$ of the vertices of $G$ whose first term is $v$. We establish a formula for the traceable number $t(v)$ of a vertex $v$ in a tree. The Hamiltonian-connected number $\mathop {\mathrm hcon}(G)$ of a connected graph $G$ is defined by $\mathop {\mathrm hcon}(G) = \sum _{v \in V(G)} t(v).$ We establish sharp bounds for $\mathop {\mathrm hcon}(G)$ of a connected graph $G$ in terms of its order.
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