• Vertex Pancyclicity and New Sufficient Conditions

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/122/03/0319-0328

• # Keywords

Hamiltonian graphs; vertex pancyclic; degree sum; neighborhood union; sufficient conditions.

• # Abstract

For a graph $G,\delta(G)$ denotes the minimum degree of 𝐺. In 1971, Bondy proved that, if 𝐺 is a 2-connected graph of order 𝑛 and $d(x)+d(y)\geq n$ for each pair of non-adjacent vertices $x,y$ in 𝐺, then 𝐺 is pancyclic or $G=K_{n/2,n/2}$. In 2001, $Xu$ proved that, if 𝐺 is a 2-connected graph of order $n\geq 6$ and $|N(x)\cup N(y)|+\delta(G)\geq n$ for each pair of non-adjacent vertices $x,y$ in 𝐺, then 𝐺 is pancyclic or $G=K_{n/2,n/2}$. In this paper, we introduce a new sufficient condition of generalizing degree sum and neighborhood union and prove that, if 𝐺 is a 2-connected graph of order $n\geq 6$ and $|N(x)\cup N(y)|+d(w)\geq n$ for any three vertices $x,y,w$ of $d(x,y)=2$ and $wx$ or $wy \notin E(G)$ in 𝐺, then 𝐺 is 4-vertex pancyclic or 𝐺 belongs to two classes of well-structured exceptional graphs. This result also generalizes the above results.

• # Author Affiliations

1. Department of Mathematics, Qiongzhou University, Sanya, Hainan 572022, People’s Republic of China

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019