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The upper traceable number of a graph. (English) Zbl 1174.05040
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:
 05C12 Distance in graphs 05C45 Eulerian and Hamiltonian graphs
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References:
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