On upper traceable numbers of graphs

Okamoto, Futaba ; Zhang, Ping

Mathematica Bohemica, Tome 133 (2008) no. 4, pp. 389-405.

Résumé

For a connected graph $G$ of order $n\ge 2$ and a linear ordering $s:v_1,v_2,\dots ,v_n$ 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.

Détail

DOI : 10.21136/MB.2008.140628

Classification : 05C12, 05C45
Mots-clés : traceable graph; traceable number; upper traceable number