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The upper traceable number of a graph

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Abstract

For a nontrivial connected graph G of order n and a linear ordering s: v 1, v 2, …, v n of vertices of G, define \( d(s) = \sum\limits_{i = 1}^{n - 1} {d(v_i ,v_{i + 1} )} \). 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.

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Authors and Affiliations

  1. Mathematics Department, University of Wisconsin-La Crosse, La Crosse, WI, 54601, USA

    Futaba Okamoto

  2. Department of Mathematics, Srinakharinwirot University, Sukhumvit Soi 23, Bangkok, 10110, Thailand

    Varaporn Saenpholphat

  3. Department of Mathematics, Western Michigan University, Kalamazoo, MI, 49008, USA

    Ping Zhang

Authors
  1. Futaba Okamoto
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  2. Ping Zhang
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  3. Varaporn Saenpholphat
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Additional information

Research supported by Srinakharinwirot University, the Thailand Research Fund and the Commission on Higher Education, Thailand under the grant number MRG 5080075.

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Okamoto, F., Zhang, P. & Saenpholphat, V. The upper traceable number of a graph. Czech Math J 58, 271–287 (2008). https://doi.org/10.1007/s10587-008-0016-9

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  • DOI: https://doi.org/10.1007/s10587-008-0016-9

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