Measures of traceability in graphs

Varaporn Saenpholphat; Futaba Okamoto; Ping Zhang

Mathematica Bohemica (2006)

  • Volume: 131, Issue: 1, page 63-84
  • ISSN: 0862-7959

Abstract

top
For a connected graph G of order n ≥ 3 and an ordering s v 1 , v 2 , ⋯ , v n of the vertices of G , d ( s ) = ∑ 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 d ( s ) , 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 2 n - 2 - p ≤ t ( G ) ≤ 2 n - 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 ≥ 3 , then t ( G ) ≤ 2 n - 4 . We present characterizations of connected graphs of order n having traceable number 2 n - 4 or 2 n - 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 { d ( s ) } , 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 h c o n ( G ) of a connected graph G is defined by h c o n ( G ) = ∑ v ∈ V ( G ) t ( v ) . We establish sharp bounds for h c o n ( G ) of a connected graph G in terms of its order.

How to cite

top

Saenpholphat, Varaporn, Okamoto, Futaba, and Zhang, Ping. "Measures of traceability in graphs." Mathematica Bohemica 131.1 (2006): 63-84. <http://eudml.org/doc/32708>.

@article{Saenpholphat2006,
abstract = {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 $\mathop \{\mathrm \{h\}con\}(G)$ of a connected graph $G$ is defined by $\mathop \{\mathrm \{h\}con\}(G) = \sum _\{v \in V(G)\} t(v).$ We establish sharp bounds for $\mathop \{\mathrm \{h\}con\}(G)$ of a connected graph $G$ in terms of its order.},
author = {Saenpholphat, Varaporn, Okamoto, Futaba, Zhang, Ping},
journal = {Mathematica Bohemica},
keywords = {traceable graph; Hamiltonian graph; Hamiltonian-connected graph; traceable graph; Hamiltonian graph; Hamiltonian-connected graph; distance; traceable number; bounds; characterizations; Hamiltonian number},
language = {eng},
number = {1},
pages = {63-84},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Measures of traceability in graphs},
url = {http://eudml.org/doc/32708},
volume = {131},
year = {2006},
}

TY - JOUR
AU - Saenpholphat, Varaporn
AU - Okamoto, Futaba
AU - Zhang, Ping
TI - Measures of traceability in graphs
JO - Mathematica Bohemica
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 131
IS - 1
SP - 63
EP - 84
AB - 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 $\mathop {\mathrm {h}con}(G)$ of a connected graph $G$ is defined by $\mathop {\mathrm {h}con}(G) = \sum _{v \in V(G)} t(v).$ We establish sharp bounds for $\mathop {\mathrm {h}con}(G)$ of a connected graph $G$ in terms of its order.
LA - eng
KW - traceable graph; Hamiltonian graph; Hamiltonian-connected graph; traceable graph; Hamiltonian graph; Hamiltonian-connected graph; distance; traceable number; bounds; characterizations; Hamiltonian number
UR - http://eudml.org/doc/32708
ER -

References

top
  1. 10.1002/jgt.3190040310, J. Graph Theory 4 (1980), 315–336. (1980) MR0584677DOI10.1002/jgt.3190040310
  2. 10.1016/0166-218X(83)90042-2, Discrete Appl. Math. 5 (1983), 211–222. (1983) MR0683513DOI10.1016/0166-218X(83)90042-2
  3. On Hamiltonian walks, Congr. Numerantium 15 (1976), 41–51. (1976) Zbl0329.05113MR0398891
  4. On the Hamiltonian number of a graph, Congr. Numerantium 165 (2003), 51–64. (2003) MR2049121
  5. A new look at Hamiltonian walks, Bull. Inst. Combin. Appl. 42 (2004), 37–52. (2004) MR2082480
  6. Introduction to Graph Theory, McGraw-Hill, Boston, 2005. (2005) 
  7. On Hamiltonian walks in graphs, Congr. Numerantium (1973), 335–342. (1973) MR0357223
  8. 10.1137/0203017, SIAM J. Comput. 3 (1974), 214–221. (1974) MR0432492DOI10.1137/0203017
  9. A generalization of Hamiltonian cycles for trees, Czech. Math. J. 26 (1976), 596–603. (1976) MR0543670
  10. 10.1007/BF02017925, Period. Math. Hungar 6 (1975), 287–293. (1975) Zbl0363.05053MR0406872DOI10.1007/BF02017925
  11. On open Hamiltonian walks in graphs, Arch. Math., Brno 27A (1991), 105–111. (1991) Zbl0758.05067MR1189647

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.