21,211 research outputs found
Dynamic Algorithms for Graph Coloring
We design fast dynamic algorithms for proper vertex and edge colorings in a
graph undergoing edge insertions and deletions. In the static setting, there
are simple linear time algorithms for - vertex coloring and
-edge coloring in a graph with maximum degree . It is
natural to ask if we can efficiently maintain such colorings in the dynamic
setting as well. We get the following three results. (1) We present a
randomized algorithm which maintains a -vertex coloring with
expected amortized update time. (2) We present a deterministic
algorithm which maintains a -vertex coloring with
amortized update time. (3) We present a simple,
deterministic algorithm which maintains a -edge coloring with
worst-case update time. This improves the recent
-edge coloring algorithm with worst-case
update time by Barenboim and Maimon.Comment: To appear in SODA 201
Upper bounds for the 2-hued chromatic number of graphs in terms of the independence number
A 2-hued coloring of a graph (also known as conditional -coloring
and dynamic coloring) is a coloring such that for every vertex of
degree at least , the neighbors of receive at least colors. The
smallest integer such that has a 2-hued coloring with colors, is
called the {\it 2-hued chromatic number} of and denoted by . In
this paper, we will show that if is a regular graph, then and if is a graph and
, then and in general case if is a graph, then .Comment: Dynamic chromatic number; conditional (k, 2)-coloring; 2-hued
chromatic number; 2-hued coloring; Independence number; Probabilistic metho
On r-Dynamic Chromatic Number of the Corronation of Path and Several Graphs
This study is a natural extension of k-proper coloring of any simple and connected graph G. By an r-dynamic coloring of a graph G, we mean a proper k-coloring of graph G such that the neighbors of any vertex v receive at least min{r, d(v)} different colors. The r-dynamic chromatic number, written as r(G), is the minimum k such that graph G has an r-dynamic k-coloring. In this paper we will study the r-dynamic chromatic number of the coronation of path and several graph. We denote the corona product of G and H by Gβ¨βH. We will obtain the r-dynamic chromatic number of Ο_r (P_nβ¨P_m ),Ο_r (P_nβ¨C_m )"and " Ο_r (P_nβ¨W_m ) for m, n>= 3
Dynamic Chromatic Number of Regular Graphs
A dynamic coloring of a graph is a proper coloring such that for every
vertex of degree at least 2, the neighbors of receive at least
2 colors. It was conjectured [B. Montgomery. {\em Dynamic coloring of graphs}.
PhD thesis, West Virginia University, 2001.] that if is a -regular
graph, then . In this paper, we prove that if is a
-regular graph with , then . It confirms the conjecture for all regular graph with
diameter at most 2 and . In fact, it shows that
provided that has diameter at most 2 and
. Moreover, we show that for any -regular graph ,
. Also, we show that for any there exists a
regular graph whose chromatic number is and .
This result gives a negative answer to a conjecture of [A. Ahadi, S. Akbari, A.
Dehghan, and M. Ghanbari. \newblock On the difference between chromatic number
and dynamic chromatic number of graphs. \newblock {\em Discrete Math.}, In
press].Comment: 8 page
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