✦ LIBER ✦

Neighborhood unions and the cycle cover number of a graph

✍ Scribed by Guantao Chen; Ronald J. Gould; Michael S. Jacobson; Richard H. Schelp

Publisher: John Wiley and Sons
Year: 1994
Tongue: English
Weight: 413 KB
Volume: 18
Category: Article
ISSN: 0364-9024
DOI: 10.1002/jgt.3190180703

No coin nor oath required. For personal study only.

✦ Synopsis

Abstract

For several years, the study of neighborhood unions of graphs has given rise to important structural consequences of graphs. In particular, neighborhood conditions that give rise to hamiltonian cycles have been considered in depth. In this paper we generalize these approaches to give a bound on the smallest number of cycles in G containing all the vertices of G.

We show that if for all x, y ≅ V(G), |N(x) ∩ N(y)| ≧ 2__n__/5 + 1, then V(G) is coverable by at most two cycles. Several related results and extensions to t cycles are also given.

📜 SIMILAR VOLUMES

Real flow number and the cycle rank of a

Real flow number and the cycle rank of a graph

✍ Robert Lukot'ka; Martin Škoviera 📂 Article 📅 2008 🏛 John Wiley and Sons 🌐 English ⚖ 101 KB

## Abstract This article establishes a relationship between the real (circular) flow number of a graph and its cycle rank. We show that a connected graph with real flow number __p__/__q__ + 1, where __p__ and __q__ are two relatively prime numbers must have cycle rank at least __p__ + __q__ − 1. A

The chromatic covering number of a graph

✍ Reza Naserasr; Claude Tardif 📂 Article 📅 2006 🏛 John Wiley and Sons 🌐 English ⚖ 72 KB 👁 2 views

Following [1] , we investigate the problem of covering a graph G with induced subgraphs G 1 ; . . . ; G k of possibly smaller chromatic number, but such that for every vertex u of G, the sum of reciprocals of the chromatic numbers of the G i 's containing u is at least 1. The existence of such ''ch

The number of shortest cycles and the ch

The number of shortest cycles and the chromatic uniqueness of a graph

✍ C. P. Teo; K. M. Koh 📂 Article 📅 1992 🏛 John Wiley and Sons 🌐 English ⚖ 403 KB 👁 2 views

## Abstract For a graph __G__, let __g__(__G__) and σ~g~(__G__) denote, respectively, the girth of __G__ and the number of cycles of length __g__(__G__) in __G__. In this paper, we first obtain an upper bound for σ~g~(__G__) and determine the structure of a 2‐connected graph __G__ when σ~g~(__G__)

On the maximum number of cycles in a pla

On the maximum number of cycles in a planar graph

✍ R. E. L. Aldred; Carsten Thomassen 📂 Article 📅 2008 🏛 John Wiley and Sons 🌐 English ⚖ 142 KB 👁 2 views

## Abstract Let __G__ be a graph on __p__ vertices with __q__ edges and let __r__ = __q__ − __p__ = 1. We show that __G__ has at most ${15\over 16} 2^{r}$ cycles. We also show that if __G__ is planar, then __G__ has at most 2^__r__ − 1^ = __o__(2^__r__ − 1^) cycles. The planar result is best possib

Covering the vertices of a graph by cycl

Covering the vertices of a graph by cycles of prescribed length

✍ D. Amar; I. Fournier; A. Germa 📂 Article 📅 1989 🏛 John Wiley and Sons 🌐 English ⚖ 321 KB

The main theorem of that paper is the following: let G be a graph of order n, of size at least (nZ -3n + 6 ) / 2 . For any integers k, n,, n2,. . . , nk such that n = n, + n2 + ... + nk and n, 2 3, there exists a covering of the vertices of G by disjoint cycles (C,),=,..,k with ICjl = n,, except whe

On the number of hamiltonian cycles in a

On the number of hamiltonian cycles in a maximal planar graph

✍ S. L. Hakimi; E. F. Schmeichel; C. Thomassen 📂 Article 📅 1979 🏛 John Wiley and Sons 🌐 English ⚖ 243 KB 👁 2 views

## Abstract We consider the problem of the minimum number of Hamiltonian cycles that could be present in a Hamiltonian maximal planar graph on __p__ vertices. In particular, we construct a __p__‐vertex maximal planar graph containing exactly four Hamiltonian cycles for every __p__ ≥ 12. We also pro