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research
On bounding the difference between the maximum degree and the chromatic number by a constant
Authors
Oliver Schaudt
Vera Weil
Publication date
13 September 2016
Publisher
Doi
Cite
View
on
arXiv
Abstract
We provide a finite forbidden induced subgraph characterization for the graph class
Î¥
k
\varUpsilon_k
Î¥
k
​
, for all
k
∈
N
0
k \in \mathbb{N}_0
k
∈
N
0
​
, which is defined as follows. A graph is in
Î¥
k
\varUpsilon_k
Î¥
k
​
if for any induced subgraph,
Δ
≤
χ
−
1
+
k
\Delta \leq \chi -1 + k
Δ
≤
χ
−
1
+
k
holds, where
Δ
\Delta
Δ
is the maximum degree and
χ
\chi
χ
is the chromatic number of the subgraph. We compare these results with those given in [O. Schaudt, V. Weil, On bounding the difference between the maximum degree and the clique number, Graphs and Combinatorics 31(5), 1689-1702 (2015). DOI: 10.1007/s00373-014-1468-3], where we studied the graph class
Ω
k
\varOmega_k
Ω
k
​
, for
k
∈
N
0
k \in \mathbb{N}_0
k
∈
N
0
​
, whose graphs are such that for any induced subgraph,
Δ
≤
ω
−
1
+
k
\Delta \leq \omega -1 + k
Δ
≤
ω
−
1
+
k
holds, where
ω
\omega
ω
denotes the clique number of a graph. In particular, we give a characterization in terms of
Ω
k
\varOmega_k
Ω
k
​
and
Î¥
k
\varUpsilon_k
Î¥
k
​
of those graphs where the neighborhood of every vertex is perfect.Comment: 10 pages, 4 figure
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