Induced path factors of regular graphs

• Published in: Journal of graph theory Vol. 97; no. 2; pp. 260 - 280
• Main Authors: Akbari, Saieed, Horsley, Daniel, Wanless, Ian M.
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01-06-2021
• Subjects: Apexes; covering; Graph theory; Graphs; induced path; path factor; regular graph; subcubic graph
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.22654

An induced path factor of a graph
G is a set of induced paths in
G with the property that every vertex of
G is in exactly one of the paths. The induced path number
ρ
(
G
) of
G is the minimum number of paths in an induced path factor of
G. We show that if
G is a connected cubic graph on
n
>
6 vertices, then
ρ
(
G
)
⩽
(
n
−
1
)
/
3. Fix an integer
k
⩾
3. For each
n, define
ℳ
n to be the maximum value of
ρ
(
G
) over all connected
k‐regular graphs
G on
n vertices. As
n
→
∞ with
n
k even, we show that
c
k
=
lim
(
ℳ
n
/
n
) exists. We prove that
5
/
18
⩽
c
3
⩽
1
/
3 and
3
/
7
⩽
c
4
⩽
1
/
2 and that
c
k
=
1
2
−
O
(
k
−
1
) for
k
→
∞.