Long paths and cycles passing through specified vertices under the average degree condition

TL;DR

This paper establishes new results on the existence of long paths and cycles passing through specified vertices in k-connected graphs, generalizing classical theorems using average degree conditions.

Contribution

It proves that in k-connected graphs, long paths and cycles passing through specified vertices exist under average degree constraints, extending prior fundamental theorems.

Findings

Existence of long paths passing through specified vertices in k-connected graphs.

Existence of long cycles passing through specified vertices based on average degree.

Generalization of Fan's and Erdős-Gallai's theorems on long paths and cycles.

Abstract

Let $G$ be a $k$-connected graph with $k\geq 2$. In this paper we first prove that: For two distinct vertices $x$ and $z$ in $G$, it contains a path passing through its any $k-2$ {specified} vertices with length at least the average degree of the vertices other than $x$ and $z$. Further, with this result, we prove that: If $G$ has $n$ vertices and $m$ edges, then it contains a cycle of length at least $2m/(n-1)$ passing through its any $k-1$ specified vertices. Our results generalize a theorem of Fan on the existence of long paths and a classical theorem of Erd\"os and Gallai on the existence of long cycles under the average degree condition.