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
Even cycles with prescribed chords in planar cubic graphs
โ Scribed by Herbert Fleischner
- Publisher
- Elsevier Science
- Year
- 1983
- Tongue
- English
- Weight
- 254 KB
- Volume
- 44
- Category
- Article
- ISSN
- 0012-365X
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โฆ Synopsis
The following result is being proved. Theorem: Let e be an arbitrary line of the 2-connected, 3-regular, planar graph G such that e cioes not belong to any cut set of size 2. Then G contains an even cycle for which e is a chord.
๐ SIMILAR VOLUMES
## Abstract Let $n\_1,n\_2,\ldots,n\_k$ be integers, $n=\sum n\_i$, $n\_i\ge 3$, and let for each $1\le i\le k$, $H\_i$ be a cycle or a tree on $n\_i$ vertices. We prove that every graph __G__ of order at least __n__ with $\sigma\_2(G) \ge 2( n-k) -1$ contains __k__ vertex disjoint subgraphs $H\_1'
A cycle C of a graph G is a D~-cycle if every component of G-V(C) has order less than 2. Using the notion of D~-cycles, a number of results are established concerning long cycles in graphs with prescribed toughness and minimum degree. Let G be a t-tough graph on n/> 3 vertices. If 6 > n/(t + 2) + 2-