学术报告

报告人： 陈冠涛教授
(Department of Mathematics and Statistics Georgia State University)
报告题目：The El-Zahar Conjecture and the Surrounding Problems
摘 要： In 1984, El-Zahar made the following conjecture.

Conjecture1:
Let $G$ is a graph of order $n=n_{1}+n_{2}$ $+ldots +n_{k}$ ($n_{i}geq 3$). If the minimum degree $$delta(G) geq lceil frac{n_{1}}{2}rceil + lceil frac{n_{2}}{2}rceil +ldots +lceil frac{n_{k}}{2}rceil, $$ then $G$ contains a $2$-factor consisting of $k$ vertex disjoint cycles of lengths $n_{1}$, $n_{2}$, $ldots $, $n_{k}$ respectively.
The case $k=1$ is the classic result of Dirac. In the same paper, El-Zahar showed the conjecture is true for $k=2$. Although the conjecture is still open, Sarmad Abbasi proved the conjecture for all sufficient large $n$ by using the Szermer'edi regularity lemma.
The El-Zahar conjecture has inspired the following two general problems.
Problem1:
Given a fixed integer $k$, when can the vertex set $V(G)$ be partitioned into $k$ vertex-disjoint cycles?

Problem2:
Suppose $n = n_1 + n_2 dots +n_k$ and $G$ is a graph of order $n$, when can $V(G)$ be partitioned into $k$ vertex-disjoint cycles of lengths $n_1$, $n_2$, $dots$, $n_k$ respectively?
On the other hand, searching for some strong structure properties which contains all possible $2$-factors has been of an interesting for long time. The works on this direction have been focused on the following two conjectures of P'osa and Seymour, respectively.

conjecture2:
Let $G$ be a graph on $n$ vertices. If $delta(G) geq frac 23n$, then $G$ contains the square of a hamiltonian cycle.

Conjecture3:
Let $G$ be a graph on $n$ vertices. If $delta(G) geqfrac{k}{k+1}n$, then $G$ contains the $k^{th}$ power of a hamiltonian cycle.
In this talk, the progresses and problems on the two problems and the two conjectures will discussed.
日 期: 2005年8月6日 上午 10:00-11:00
地 点:A1510

欢迎参加！

数学系
2005.8.4