101 research outputs found
On The b-Chromatic Number of Regular Graphs Without 4-Cycle
The b-chromatic number of a graph , denoted by , is the largest
integer that admits a proper -coloring such that each color class
has a vertex that is adjacent to at least one vertex in each of the other color
classes. We prove that for each -regular graph which contains no
4-cycle, and if has a triangle,
then . Also, if is a -regular
graph which contains no 4-cycle and , then .
Finally, we show that for any -regular graph which does not contain
4-cycle and ,
The b-Chromatic Number of Regular Graphs via The Edge Connectivity
\noindent The b-chromatic number of a graph , denoted by , is the
largest integer that admits a proper coloring by colors, such that
each color class has a vertex that is adjacent to at least one vertex in each
of the other color classes. El Sahili and Kouider [About b-colorings of regular
graphs, Res. Rep. 1432, LRI, Univ. Orsay, France, 2006] asked whether it is
true that every -regular graph of girth at least 5 satisfies
. Blidia, Maffray, and Zemir [On b-colorings in regular graphs,
Discrete Appl. Math. 157 (2009), 1787-1793] showed that the Petersen graph
provides a negative answer to this question, and then conjectured that the
Petersen graph is the only exception. In this paper, we investigate a
strengthened form of the question.
The edge connectivity of a graph , denoted by , is the minimum
cardinality of a subset of such that is either
disconnected or a graph with only one vertex. A -regular graph is called
super-edge-connected if every minimum edge-cut is the set of all edges incident
with a vertex in , i.e., and every minimum edge-cut of
isolates a vertex. We show that if is a -regular graph that contains no
4-cycle, then whenever is not super-edge-connected
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