# Article

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Keywords:
traceable graph; traceable number; upper traceable number
Summary:
For a connected graph $G$ of order $n\ge 2$ and a linear ordering $s\colon v_1,v_2,\ldots ,v_n$ of 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 upper traceable number $t^+(G)$ of $G$ is $t^+(G)= \max \{d(s)\}$, where the maximum is taken over all linear orderings $s$ of vertices of $G$. It is known that if $T$ is a tree of order $n\ge 3$, then $2n-3\le t^+(T)\le \lfloor {n^2/2}\rfloor -1$ and $t^+(T)\le \lfloor {n^2/2}\rfloor -3$ if $T\ne P_n$. All pairs $n,k$ for which there exists a tree $T$ of order $n$ and $t^+(T)= k$ are determined and a characterization of all those trees of order $n\ge 4$ with upper traceable number $\lfloor {n^2/2}\rfloor -3$ is established. For a connected graph $G$ of order $n\ge 3$, it is known that $n-1\le t^+(G)\le \lfloor {n^2/2}\rfloor -1$. We investigate the problem of determining possible pairs $n,k$ of positive integers that are realizable as the order and upper traceable number of some connected graph.
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