We propose a conjecture: for each integer k β₯ 2, there exists N (k) such that if G is a graph of order n β₯ N (k) and d(x) + d(y) β₯ n + 2k -2 for each pair of nonadjacent vertices x and y of G, then for any k independent edges e 1 , . . . , e k of G, there exist If this conjecture is true, the condi
Covering a graph with cycles
β Scribed by Hong Wang
- Publisher
- John Wiley and Sons
- Year
- 1995
- Tongue
- English
- Weight
- 444 KB
- Volume
- 20
- Category
- Article
- ISSN
- 0364-9024
No coin nor oath required. For personal study only.
β¦ Synopsis
Abstract
Let k and n be two integers such that k β₯ 0 and n β₯ 3(k + 1). Let G be a graph of order n with minimum degree at least β(n + k)/2β. Then G contains k + 1 independent cycles covering all the vertices of G such that k of them are triangles. Β© 1995, John Wiley & Sons, Inc.
π SIMILAR VOLUMES
## Abstract It is shown that the edges of a simple graph with a nowhereβzero 4βflow can be covered with cycles such that the sum of the lengths of the cycles is at most |__E__(__G__)| + |__V__(__G__)| β3. This solves a conjecture proposed by G. Fan.
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
Some new results on minimum cycle covers are proved. As a consequence, it is obtained that the edges of a bridgeless graph G can be covered by cycles of total length at most |E(G)| + 25 24 (|V (G)| -1), and at most |E(G)| + |V (G)| -1 if G contains no circuit of length 8 or 12.
In this article, we show that for any simple, bridgeless graph G on n vertices, there is a family C of at most n-1 cycles which cover the edges of G at least twice. A similar, dual result is also proven for cocycles namely: for any loopless graph G on n vertices and edges having cogirth g \* β₯ 3 and
## Abstract Let __G__ be a bridgeless cubic graph. We prove that the edges of __G__ can be covered by circuits whose total length is at most (44/27) |__E(G)__|, and if Tutte's 3βflow Conjecture is true, at most (92/57) |__E(G)__|.