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Journal Articles Discrete Mathematics and Theoretical Computer Science Year : 2016

## On degree-sequence characterization and the extremal number of edges for various Hamiltonian properties under fault tolerance

Shih-Yan Chen
• Function : Author
Hsun Su
• Function : Author

#### Abstract

Assume that $n, \delta ,k$ are integers with $0 \leq k < \delta < n$. Given a graph $G=(V,E)$ with $|V|=n$. The symbol $G-F, F \subseteq V$, denotes the graph with $V(G-F)=V-F$, and $E(G-F)$ obtained by $E$ after deleting the edges with at least one endvertex in $F$. $G$ is called $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, or $k$-vertex fault Hamiltonian-connected if $G-F$ remains traceable, Hamiltonian, and Hamiltonian-connected for all $F$ with $0 \leq |F| \leq k$, respectively. The notations $h_1(n, \delta ,k)$, $h_2(n, \delta ,k)$, and $h_3(n, \delta ,k)$ denote the minimum number of edges required to guarantee an $n$-vertex graph with minimum degree $\delta (G) \geq \delta$ to be $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, and $k$-vertex fault Hamiltonian-connected, respectively. In this paper, we establish a theorem which uses the degree sequence of a given graph to characterize the $k$-vertex fault traceability/hamiltonicity/Hamiltonian-connectivity, respectively. Then we use this theorem to obtain the formulas for $h_i(n, \delta ,k)$ for $1 \leq i \leq 3$, which improves and extends the known results for $k=0$.

#### Domains

Discrete Mathematics [cs.DM]

### Dates and versions

hal-01352843 , version 1 (17-08-2016)

### Identifiers

• HAL Id : hal-01352843 , version 1
• DOI :

### Cite

Shih-Yan Chen, Shin-Shin Kao, Hsun Su. On degree-sequence characterization and the extremal number of edges for various Hamiltonian properties under fault tolerance. Discrete Mathematics and Theoretical Computer Science, 2016, Vol. 17 no. 3 (3), pp.307-314. ⟨10.46298/dmtcs.2149⟩. ⟨hal-01352843⟩

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