2 research outputs found
The b-continuity of graphs with large girth
A b-coloring of the vertices of a graph is a proper coloring where each color
class contains a vertex which is adjacent to each other color class. The
b-chromatic number of is the maximum integer for which has a
b-coloring with colors. A graph is b-continuous if has a
b-coloring with colors, for every integer in the interval
. It is known that not all graphs are b-continuous. In this
article, we show that if has girth at least 10, then is b-continuous.Comment: 10 page
b-continuity and Partial Grundy Coloring of graphs with large girth
A b-coloring of a graph is a proper coloring such that each color class has
at least one vertex which is adjacent to each other color class. The b-spectrum
of is the set of integers such that has a b-coloring
with colors and is the b-chromatic number of . A
graph is b-continous if . An infinite
number of graphs that are not b-continuous is known. It is also known that
graphs with girth at least 10 are b-continuous.
A partial Grundy coloring is a proper coloring such that each color class contains some vertex that is
adjacent to every color class such that . The partial Grundy number of
is the maximum value for which has a partial Grundy
coloring.
In this work, we prove that graphs with girth at least 8 are b-continuous,
and that the b-spectrum of a graph with girth at least 7 contains the
integers between and . We also prove that
equals a known upper bound when is a graph with girth at least 7. These
results generalize previous ones by Linhares-Sales and Silva (2017), and by Shi
et al.(2005).Comment: 11 pages; 4 figures; partially presented at LAGOS'1