# 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.
References:
[1] Asano, T., Nishizeki, T., Watanabe, T.: An upper bound on the length of a Hamiltonian walk of a maximal planar graph. J. Graph Theory 4 (1980), 315-336. MR 0584677 | Zbl 0433.05037
[2] Asano, T., Nishizeki, T., Watanabe, T.: An approximation algorithm for the Hamiltonian walk problems on maximal planar graphs. Discrete Appl. Math. 5 (1983), 211-222. MR 0683513
[3] Bermond, J. C.: On Hamiltonian walks. Congr. Numer. 15 (1976), 41-51. MR 0398891 | Zbl 0329.05113
[4] Chartrand, G., Kronk, H. V.: On a special class of hamiltonian graphs. Comment. Math. Helv. 44 (1969), 84-88. MR 0239995 | Zbl 0169.55501
[5] Chartrand, G., Saenpholphat, V., Thomas, T., Zhang, P.: A new look at Hamiltonian walks. Bull. Inst. Combin. Appl. 42 (2004), 37-52. MR 2082480
[6] Chartrand, G., Zhang, P.: Introduction to Graph Theory. McGraw-Hill, Boston (2005). Zbl 1096.05001
[7] Goodman, S. E., Hedetniemi, S. T.: On Hamiltonian walks in graphs. SIAM J. Comput. 3 (1974), 214-221. MR 0432492 | Zbl 0269.05113
[8] Okamoto, F., Saenpholphat, V., Zhang, P.: Measures of traceability in graphs. Math. Bohem. 131 (2006), 63-83. MR 2211004 | Zbl 1112.05032
[9] Okamoto, F., Saenpholphat, V., Zhang, P.: The upper traceable number of a graph. (to appear) in Czech. Math. J. MR 2402537 | Zbl 1174.05040
[10] Nebeský, L.: A generalization of Hamiltonian cycles for trees. Czech. Math. J. 26 (1976), 596-603. MR 0543670
[11] Thomassen, C.: On randomly Hamiltonian graphs. Math. Ann. 200 (1973), 195-208. MR 0325456 | Zbl 0233.05121
[12] Vacek, P.: On open Hamiltonian walks in graphs. Arch. Math., Brno 27A (1991), 105-111. MR 1189647 | Zbl 0758.05067
[13] Vacek, P.: Bounds of lengths of open Hamiltonian walks. Arch. Math., Brno 28 (1992), 11-16. MR 1201861 | Zbl 0782.05056

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