On Vertex-Disjoint Triangles in Tripartite Graphs and Multigraphs

• Published in: Graphs and combinatorics Vol. 36; no. 5; pp. 1355 - 1361
• Main Authors: Zou, Qingsong, Li, Jiawang, Ji, Zizheng
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01-09-2020; Springer Nature B.V
• Subjects: Apexes; Combinatorics; Engineering Design; Graph theory; Mathematics; Mathematics and Statistics; Original Paper; Cycles; Triangles; Tripartite graphs; Multigraphs; 05C70; Vertex-disjoint; 05C38
• ISSN: 0911-0119; 1435-5914
• DOI: 10.1007/s00373-020-02188-3

Let
G
be a tripartite graph with tripartition
(
V
1
,
V
2
,
V
3
)
, where
∣
V
1
∣
=
∣
V
2
∣
=
∣
V
3
∣
=
k
>
0
. It is proved that if
d
(
x
)
+
d
(
y
)
≥
3
k
for every pair of nonadjacent vertices
x
∈
V
i
,
y
∈
V
j
with
i
≠
j
(
i
,
j
∈
{
1
,
2
,
3
}
)
, then
G
contains
k
vertex-disjoint triangles. As a corollary, if
d
(
x
)
≥
3
2
k
for each vertex
x
∈
V
(
G
)
, then
G
contains
k
vertex-disjoint triangles. Based on the above results, vertex-disjoint triangles in multigraphs are studied. Let
M
be a standard tripartite multigraph with tripartition
(
V
1
,
V
2
,
V
3
)
, where
∣
V
1
∣
=
∣
V
2
∣
=
∣
V
3
∣
=
k
>
0
. If
δ
(
M
)
≥
3
k
-
1
for even
k
and
δ
(
M
)
≥
3
k
for odd
k
, then
M
contains
k
vertex-disjoint 4-triangles
Δ
4
(a triangle with at least four edges). Furthermore, examples are given showing that the degree conditions of all our three results are best possible.