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Zbl 1174.05040
Okamoto, Futaba; Zhang, Ping; Saenpholphat, Varaporn
The upper traceable number of a graph. (English)
[J] Czech. Math. J. 58, No. 1, 271-287 (2008). ISSN 0011-4642; ISSN 1572-9141/e

Summary: For a nontrivial connected graph $G$ of order $n$ and a linear ordering $s\: v_1, v_2, \ldots , v_n$ of vertices of $G$, define $d(s) = \sum _{i=1}^{n-1} d(v_i, v_{i+1})$. The traceable number $t(G)$ of a graph $G$ is $t(G) = \min \{d(s)\}$ and the upper traceable number $t^+(G)$ of $G$ is $t^+(G) = \max \{d(s)\},$ where the minimum and maximum are taken over all linear orderings $s$ of vertices of $G$. We study upper traceable numbers of several classes of graphs and the relationship between the traceable number and upper traceable number of a graph. All connected graphs $G$ for which $t^+(G)- t(G) = 1$ are characterized and a formula for the upper traceable number of a tree is established.

MSC 2000:: *05C12 Distance in graphs
05C45 Eulerian and Hamiltonian graphs

Keywords: traceable number; upper traceable number; Hamiltonian number

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