hamiltonian connected
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2021 ◽  
Vol 37 (4) ◽  
pp. 800-806
Author(s):  
Ren-sen Ma ◽  
Ai-mei Yu ◽  
Ke-ke Wang ◽  
Hong-Jian Lai

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Ayesha Shabbir ◽  
Muhammad Faisal Nadeem ◽  
Tudor Zamfirescu

A spanning path in a graph G is called a Hamiltonian path. To determine which graphs possess such paths is an NP-complete problem. A graph G is called Hamiltonian-connected if any two vertices of G are connected by a Hamiltonian path. We consider here the family of Toeplitz graphs. About them, it is known only for n=3 that Tnp,q is Hamiltonian-connected, while some particular cases of Tnp,q,r for p=1 and q=2,3,4 have also been investigated regarding Hamiltonian connectedness. Here, we prove that the nonbipartite Toeplitz graph Tn1,q,r is Hamiltonian-connected for all 1<q<r<n and n≥5r−2.


2018 ◽  
Vol 3 (2) ◽  
pp. 419-426 ◽  
Author(s):  
Donglin Liu ◽  
Chunxiang Wang ◽  
Shaohui Wang

AbstractThe product graph Gm *Gp of two given graphs Gm and Gp, defined by J.C. Bermond et al.[J Combin Theory, Series B 36(1984) 32-48] in the context of the so-called (Δ,D)-problem, is one interesting model in the design of large reliable networks. This work deals with sufficient conditions that guarantee these product graphs to be hamiltonian-connected. Moreover, we state product graphs for which provide panconnectivity of interconnection networks modeled by a product of graphs with faulty elements.


2018 ◽  
Vol 341 (5) ◽  
pp. 1363-1379 ◽  
Author(s):  
Jianping Liu ◽  
Aimei Yu ◽  
Keke Wang ◽  
Hong-Jian Lai

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Shih-Yan Chen ◽  
Shin-Shin Kao ◽  
Hsun Su

International audience 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 <i>$k$-vertex fault traceable</i>, <i>$k$-vertex fault Hamiltonian</i>, or <i>$k$-vertex fault Hamiltonian-connected</i> 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 $k=0$.


2016 ◽  
Vol 30 (1) ◽  
pp. 112-140 ◽  
Author(s):  
Ken-ichi Kawarabayashi ◽  
Kenta Ozeki

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