Short cycle covers of cubic graphs
β Scribed by Genghua Fan
- Publisher
- John Wiley and Sons
- Year
- 1994
- Tongue
- English
- Weight
- 461 KB
- Volume
- 18
- Category
- Article
- ISSN
- 0364-9024
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β¦ Synopsis
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)|.
π SIMILAR VOLUMES
## Abstract Let __SCC__~3~(__G__) be the length of a shortest 3βcycle cover of a bridgeless cubic graph __G__. It is proved in this note that if __G__ contains no circuit of length 5 (an improvement of Jackson's (__JCTB 1994__) result: if __G__ has girth at least 7) and if all 5βcircuits of __G_
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.
We show that the edge set of a bridgeless cubic graph \(G\) can be covered with circuits such that the sum of the lengths of the circuits is at most \(\frac{64}{39}|E(G)|\). Stronger results are obtained for cubic graphs of large girth. 1994 Academic Press, Inc.
We describe a general sufficient condition for a Hamiltonian graph to contain another Hamiltonian cycle. We apply it to prove that every longest cycle in a 3-connected cubic graph has a chord. We also verify special cases of an old conjecture of Sheehan on Hamiltonian cycles in 4-regular graphs and