Geodesic
A Note on Cycles in Locally Hamiltonian and Locally Hamilton-Connected Graphs
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 77-84

Voir la notice de l'article provenant de la source Library of Science

Let 𝒫 be a property of a graph. A graph G is said to be locally 𝒫, if the subgraph induced by the open neighbourhood of every vertex in G has property 𝒫. Ryjáček conjectures that every connected, locally connected graph is weakly pancyclic. Motivated by the above conjecture, van Aardt et al. [S.A.van Aardt, M. Frick, O.R. Oellermann and J.P.de Wet, Global cycle properties in locally connected, locally traceable and locally Hamiltonian graphs, Discrete Appl. Math. 205 (2016) 171–179] investigated the global cycle structures in connected, locally traceable/Hamiltonian graphs. Among other results, they proved that a connected, locally Hamiltonian graph G with maximum degree at least |V (G)| − 5 is weakly pancyclic. In this note, we improve this result by showing that such a graph with maximum degree at least |V (G)|−6 is weakly pancyclic. Furthermore, we show that a connected, locally Hamilton-connected graph with maximum degree at most 7 is fully cycle extendable.
Keywords: locally connected, locally Hamiltonian, locally Hamilton-connected, fully cycle extendability, weakly pancyclicity 
@article{DMGT_2020_40_1_a5,
     author = {Tang, Long and Vumar, Elkin},
     title = {A {Note} on {Cycles} in {Locally} {Hamiltonian} and {Locally} {Hamilton-Connected} {Graphs}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {77--84},
     publisher = {mathdoc},
     volume = {40},
     number = {1},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a5/}
}
TY  - JOUR
AU  - Tang, Long
AU  - Vumar, Elkin
TI  - A Note on Cycles in Locally Hamiltonian and Locally Hamilton-Connected Graphs
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2020
SP  - 77
EP  - 84
VL  - 40
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a5/
LA  - en
ID  - DMGT_2020_40_1_a5
ER  - 
%0 Journal Article
%A Tang, Long
%A Vumar, Elkin
%T A Note on Cycles in Locally Hamiltonian and Locally Hamilton-Connected Graphs
%J Discussiones Mathematicae. Graph Theory
%D 2020
%P 77-84
%V 40
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a5/
%G en
%F DMGT_2020_40_1_a5
Tang, Long; Vumar, Elkin. A Note on Cycles in Locally Hamiltonian and Locally Hamilton-Connected Graphs. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 77-84. http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a5/