## Involve: A Journal of Mathematics

• Involve
• Volume 2, Number 1 (2009), 95-114.

### Hamiltonian labelings of graphs

#### Abstract

For a connected graph $G$ of order $n$, the detour distance $D(u,v)$ between two vertices $u$ and $v$ in $G$ is the length of a longest $u−v$ path in $G$. A Hamiltonian labeling of $G$ is a function $c:V(G)→ℕ$ such that $|c(u)−c(v)|+D(u,v)≥n$ for every two distinct vertices $u$ and $v$ of $G$. The value $hn(c)$ of a Hamiltonian labeling $c$ of $G$ is the maximum label (functional value) assigned to a vertex of $G$ by $c$; while the Hamiltonian labeling number $hn(G)$ of $G$ is the minimum value of Hamiltonian labelings of $G$. Hamiltonian labeling numbers of some well-known classes of graphs are determined. Sharp upper and lower bounds are established for the Hamiltonian labeling number of a connected graph. The corona $cor(F)$ of a graph $F$ is the graph obtained from $F$ by adding exactly one pendant edge at each vertex of $F$. For each integer $k≥3$, let $ℋk$ be the set of connected graphs $G$ for which there exists a Hamiltonian graph $H$ of order $k$ such that $H⊂G⊆ cor(H)$. It is shown that $2k−1≤ hn(G)≤k(2k−1)$ for each $G∈ℋk$ and that both bounds are sharp.

#### Article information

Source
Involve, Volume 2, Number 1 (2009), 95-114.

Dates
Accepted: 15 November 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.involve/1513799120

Digital Object Identifier
doi:10.2140/involve.2009.2.95

Mathematical Reviews number (MathSciNet)
MR2501348

Zentralblatt MATH identifier
1167.05024

#### Citation

Renzema, Willem; Zhang, Ping. Hamiltonian labelings of graphs. Involve 2 (2009), no. 1, 95--114. doi:10.2140/involve.2009.2.95. https://projecteuclid.org/euclid.involve/1513799120

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