The Tight Bound for the Strong Chromatic Indices of Claw-Free Subcubic Graphs
Let G be a graph and k a positive integer. A strong k -edge-coloring of G is a mapping ϕ : E ( G ) → { 1 , 2 , ⋯ , k } such that for any two edges e and e ′ that are either adjacent to each other or adjacent to a common edge, ϕ ( e ) ≠ ϕ ( e ′ ) . The strong chromatic index of G , denoted as χ s ′ (...
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| Published in: | Graphs and combinatorics Vol. 39; no. 3; p. 58 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Tokyo
Springer Japan
01.05.2023
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0911-0119, 1435-5914 |
| Online Access: | Get full text |
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| Summary: | Let
G
be a graph and
k
a positive integer. A strong
k
-edge-coloring of
G
is a mapping
ϕ
:
E
(
G
)
→
{
1
,
2
,
⋯
,
k
}
such that for any two edges
e
and
e
′
that are either adjacent to each other or adjacent to a common edge,
ϕ
(
e
)
≠
ϕ
(
e
′
)
. The strong chromatic index of
G
, denoted as
χ
s
′
(
G
)
, is the minimum integer
k
such that
G
has a strong
k
-edge-coloring. Lv, Li and Zhang [Graphs and Combinatorics 38 (3) (2022) 63] proved that if
G
is a claw-free subcubic graph other than the triangular prism then
χ
s
′
(
G
)
≤
8
. In addition, they asked if the upper bound 8 can be improved to 7. In this paper, we answer this question in the affirmative. Our proof implies a polynomial-time algorithm for finding strong 7-edge-colorings of such graphs. We also construct infinitely many claw-free subcubic graphs with their strong chromatic indices attaining the bound 7. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0911-0119 1435-5914 |
| DOI: | 10.1007/s00373-023-02655-7 |