| Title:
|
Measures of traceability in graphs (English) |
| Author:
|
Saenpholphat, Varaporn |
| Author:
|
Okamoto, Futaba |
| Author:
|
Zhang, Ping |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
131 |
| Issue:
|
1 |
| Year:
|
2006 |
| Pages:
|
63-84 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For a connected graph $G$ of order $n \ge 3$ and an ordering $s\: v_1$, $v_2, \cdots , v_n$ of the 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 traceable number $t(G)$ of $G$ is defined by $t(G) = \min \left\rbrace d(s)\right\lbrace ,$ where the minimum is taken over all sequences $s$ of the elements of $V(G)$. It is shown that if $G$ is a nontrivial connected graph of order $n$ such that $l$ is the length of a longest path in $G$ and $p$ is the maximum size of a spanning linear forest in $G$, then $2n-2 - p \le t(G) \le 2n-2 - l$ and both these bounds are sharp. We establish a formula for the traceable number of every tree in terms of its order and diameter. It is shown that if $G$ is a connected graph of order $n \ge 3$, then $t(G)\le 2n-4$. We present characterizations of connected graphs of order $n$ having traceable number $2n-4$ or $2n-5$. The relationship between the traceable number and the Hamiltonian number (the minimum length of a closed spanning walk) of a connected graph is studied. The traceable number $t(v)$ of a vertex $v$ in a connected graph $G$ is defined by $t(v) = \min \lbrace d(s)\rbrace $, where the minimum is taken over all linear orderings $s$ of the vertices of $G$ whose first term is $v$. We establish a formula for the traceable number $t(v)$ of a vertex $v$ in a tree. The Hamiltonian-connected number $\mathop {\mathrm hcon}(G)$ of a connected graph $G$ is defined by $\mathop {\mathrm hcon}(G) = \sum _{v \in V(G)} t(v).$ We establish sharp bounds for $\mathop {\mathrm hcon}(G)$ of a connected graph $G$ in terms of its order. (English) |
| Keyword:
|
traceable graph |
| Keyword:
|
Hamiltonian graph |
| Keyword:
|
Hamiltonian-connected graph |
| MSC:
|
05C12 |
| MSC:
|
05C45 |
| idZBL:
|
Zbl 1112.05032 |
| idMR:
|
MR2211004 |
| DOI:
|
10.21136/MB.2006.134076 |
| . |
| Date available:
|
2009-09-24T22:24:19Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134076 |
| . |
| Reference:
|
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