A Tight Linear Bound to the Chromatic Number of (P5,K1+(K1∪K3))-Free Graphs
Let F 1 and F 2 be two disjoint graphs. The union F 1 ∪ F 2 is a graph with vertex set V ( F 1 ) ∪ V ( F 2 ) and edge set E ( F 1 ) ∪ E ( F 2 ) , and the join F 1 + F 2 is a graph with vertex set V ( F 1 ) ∪ V ( F 2 ) and edge set E ( F 1 ) ∪ E ( F 2 ) ∪ { x y | x ∈ V ( F 1 ) and y ∈ V ( F 2 ) } . I...
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| Veröffentlicht in: | Graphs and combinatorics 2023-06, Vol.39 (3), Article 43 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | Let
F
1
and
F
2
be two disjoint graphs. The union
F
1
∪
F
2
is a graph with vertex set
V
(
F
1
)
∪
V
(
F
2
)
and edge set
E
(
F
1
)
∪
E
(
F
2
)
, and the join
F
1
+
F
2
is a graph with vertex set
V
(
F
1
)
∪
V
(
F
2
)
and edge set
E
(
F
1
)
∪
E
(
F
2
)
∪
{
x
y
|
x
∈
V
(
F
1
)
and
y
∈
V
(
F
2
)
}
. In this paper, we present a characterization to
(
P
5
,
K
1
∪
K
3
)
-free graphs, prove that
χ
(
G
)
≤
2
ω
(
G
)
-
1
if
G
is
(
P
5
,
K
1
∪
K
3
)
-free. Based on this result, we further prove that
χ
(
G
)
≤
max
{
2
ω
(
G
)
,
15
}
if
G
is a
(
P
5
,
K
1
+
(
K
1
∪
K
3
)
)
-free graph. We also construct a
(
P
5
,
K
1
+
(
K
1
∪
K
3
)
)
-free graph
G
with
χ
(
G
)
=
2
ω
(
G
)
. |
|---|---|
| ISSN: | 0911-0119 1435-5914 |
| DOI: | 10.1007/s00373-023-02642-y |