## Abstract We show that there are 2^[n/2]β4^ largest graphs of order __n__ β₯ 7 having exactly one hamiltonian cycle. a recursive procedure for constructing these graphs is described.
Extremal maximal uniquely hamiltonian graphs
β Scribed by Curtiss A. Barefoot; R. C. Entringer
- Publisher
- John Wiley and Sons
- Year
- 1980
- Tongue
- English
- Weight
- 352 KB
- Volume
- 4
- Category
- Article
- ISSN
- 0364-9024
No coin nor oath required. For personal study only.
β¦ Synopsis
Abstract
Let G be a graph of order n with exactly one Hamiltonian cycle and suppose that G is maximal with respect to this property. We determine the minimum number of edges G can have.
π SIMILAR VOLUMES
Let G be a uniquely hamiltonian graph on n vertices. We show that G has a vertex of degree at most c log 2 8n+3, where c=(2&log 2 3) &1 r2.41. We show further that G has at least two vertices of degree less than four if it is planar and at least four vertices of degree two if it is bipartite.
## Abstract A partition of the vertices of a graph is called a clumping if, for vertices in distinct partition classes, adjacency depends only on the partition classes, not on the specific vertices. We give a simple necessary and sufficient condition for a finite graph to have a unique maximal clum
## Abstract We construct 3βregular (cubic) graphs __G__ that have a dominating cycle __C__ such that no other cycle __C__~1~ of __G__ satisfies __V(C)__ β __V__(__C__~1~). By a similar construction we obtain loopless 4βregular graphs having precisely one hamiltonian cycle. The basis for these const
## Abstract The generalized Petersen graph __P__(6__k__ + 3, 2) has exactly 3 Hamiltonian cycles for __k__ β₯ 0, but for __k__ β₯ 2 is not uniquely edge colorable. This disproves a conjecture of Greenwell and Kronk [1].
## Abstract A (1,2)βeulerian weight __w__ of a grph is hamiltonian if every faithful cover of __w__ is a set of two Hamilton circuits. Let __G__ be a 3βconnected cubic graph containing no subdivition of the Petersen graph. We prove that if __G__ admits a hamiltonian weight then __G__ is uniquely 3β