Vertex Pancyclicity and New Sufficient Conditions

Proceedings Mathematical Sciences

Kewen Zhao²  Yue Lin¹ 

[1] $$;Department of Mathematics, Qiongzhou University, Sanya, Hainan 0, People’s Republic of China$$

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

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

For a graph 𝐺,𝛿(𝐺) denotes the minimum degree of 𝐺. In 1971, Bondy proved that, if 𝐺 is a 2-connected graph of order 𝑛 and $d(x)+d(y)≥ 𝑛$ 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 $n≥ 6$ and $|N(x)cup N(y)|+𝛿(G)≥ n$ 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 $n≥ 6$ and $|N(x)cup N(y)|+d(w)≥ n$ for any three vertices 𝑥,𝑦,𝑤 of $d(x,y)=2$ and 𝑤𝑥 or 𝑤𝑦 $otin E(G)$ 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.

【 授权许可】

Unknown