A look at cycles containing specified elements of a graph

AbstractThis article is intended as a brief survey of problems and results dealing with cycles containing specified elements of a graph. It is hoped that this will help researchers in the area to identify problems and areas of concentration

Cycles Through Specified Vertices In 1-Tough Graphs

. Bollob'as, Brightwell [1] and independently Shi [3] proved the existence of a cycle through all vertices of degree at least n 2 in any 2-connected graph of order n. The aim of this paper is to show that the above degree requirement can be relaxed for 1-tough graphs. 1. Introduction Bollob'as and Brightwell [1] proved that if G is a graph of order n and W is a set of w vertices of degree at least d, then there is a cycle through at least d w d n d e\Gamma1 e vertices of W . In the case d n 2 this implies the existence of a cycle through all vertices of degree at least n 2 . This special case was proved independently by Shi [3] for 2-connected graphs. In fact, Bollob'as and Brightwell proved an Ore type result which can be read as Theorem 1. [1] Let G be a graph on n vertices and let W ` VG such that each pair of non-adjacent vertices u; v 2 W satisfies d(u) + d(v) n. If jW j 3, then G contains a cycle through all vertices of W . In this paper we show that the abo..