An ILP Approach to Determine Smallest 4-Regular Non-Hamiltonian, Nontraceable, and Nonhomogeneously Traceable Graphs
In this paper we study some open questions related to the smallest order of a -regular graph which has a connectivity property but does not have a hamiltonian property . In particular, is either connectivity, -connectivity or -toughness and is hamiltonicity, homogeneous traceability or traceability....
Saved in:
| Published in: | Journal of applied and industrial mathematics Vol. 16; no. 2; pp. 252 - 266 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Moscow
Pleiades Publishing
2022
Springer Nature B.V |
| Subjects: | |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | In this paper we study some open questions related to the smallest order
of a
-regular graph which has a connectivity property
but does not have a hamiltonian property
. In particular,
is either connectivity,
-connectivity or
-toughness and
is hamiltonicity, homogeneous traceability or traceability. A standard theoretical approach to these questions had already been used in the literature, but in many cases did not succeed in determining the exact value of
. Here we have chosen to use Integer Linear Programming and to encode the graphs that we are looking for as the binary solutions to a suitable set of linear inequalities. This way, there would exist a graph of order
with certain properties if and only if the corresponding ILP had a feasible solution, which we have determined through a branch-and-cut procedure. By using our approach, we have been able to compute
for all the pairs of considered properties with the exception of
1-toughness,
traceability. Even in this last case, we have nonetheless significantly reduced the interval
in which
was known to lie. Finally, we have shown that for each
(
in the last case) there exists a
-regular graph on
vertices which has property
but not property
. |
|---|---|
| ISSN: | 1990-4789 1990-4797 |
| DOI: | 10.1134/S1990478922020077 |