**Originator(s):**
Cun-Quan Zhang, West Virginia University

**Conjecture/Question:**
Every 3-connected 3-regular graph having a Hamiltonian weight arises
from *K _{4}* by a sequence of Delta-Wye operations.

**Definitions/Background/motivation:**
A *Hamiltonian weight* on *G* is a map *f* from *E(G)*
to {1,2} such that every family of cycles that covers each edge *e*
exactly *f(e)* times consists of two Hamiltonian cycles.
A *Delta-Wye* operation replaces a triangle in a 3-regular graph
with a single vertex incident to the three edges that emanated from the
triangle.

The study of Hamiltonian weights is motivated by the cycle double cover conjecture of Szekeres and Seymour and by the unique edge-3-coloring conjecture of Fiorini and Wilson.

**Partial results:**
The conjecture was proved in [1] for those graphs not having the Petersen
graph as a minor.

**References:**

[1] Lai, Hong-Jian; Zhang, Cun-Quan.
Hamilton weights and Petersen minors.
J. Graph Theory 38 (2001), no. 4, 197--219; MR2002g:05120
05C45 (05C70)

**Keywords:**