On the dynamic coloring of graphs (original) (raw)
Related papers
2009
A dynamic coloring of a graph GGG is a proper coloring such that for every vertex vinV(G)v\in V(G)vinV(G) of degree at least 2, the neighbors of vvv receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number of graphs. In this regard, we shall show that there is a constant ccc such
On the difference between chromatic number and dynamic chromatic number of graphs
Discrete Mathematics, 2012
A proper vertex k-coloring of a graph G is called dynamic, if there is no vertex v ∈ V (G) with d(v) ≥ 2 and all of its neighbors have the same color. The smallest integer k such that G has a k-dynamic coloring is called the dynamic chromatic number of G and denoted by χ 2 (G). We say that v ∈ V (G) in a proper vertex coloring of G is a bad vertex if d(v) ≥ 2 and only one color appears in the neighbors of v. In this paper, we show that if G is a graph with the chromatic number at least 6, then there exists a proper vertex χ (G)-coloring of G such that the set of bad vertices of G is an independent set. Also, we provide some upper bounds for χ 2 (G) − χ (G) in terms of some parameters of the graph G.
On the b-chromatic number of regular graphs
Discrete Applied Mathematics, 2011
The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring in which every color class contains at least one vertex that has a neighbor in each of the other color classes. We prove that every d-regular graph with at least 2d 3 vertices has b-chromatic number d + 1, that the b-chromatic number of an arbitrary d-regular graph with girth g = 5 is at least d+1 2 and that every d-regular graph, d ≥ 6, with diameter at least d and with no 4-cycles admits a b-coloring with d + 1 colors.
Upper bounds for the 2-hued chromatic number of graphs in terms of the independence number
Discrete Applied Mathematics, 2012
A 2-hued coloring of a graph G (also known as conditional (k, 2)-coloring and dynamic coloring) is a coloring such that for every vertex v ∈ V (G) of degree at least 2, the neighbors of v receive at least 2 colors. The smallest integer k such that G has a 2-hued coloring with k colors, is called the 2-hued chromatic number of G and denoted by χ 2 (G). In this paper, we will show that if G is a regular graph, then χ 2 (G) − χ(G) ≤ 2 log 2 (α(G)) + O(1) and if G is a graph and δ(G) ≥ 2, then χ 2 (G) − χ(G) ≤ 1 + ⌈ δ−1 √ 4∆ 2 ⌉(1 + log 2∆(G) 2∆(G)−δ(G) (α(G))) and in general case if G is a graph, then χ 2 (G) − χ(G) ≤ 2 + min{α ′ (G), α(G)+ω(G) 2 }.
A new upper bound for the chromatic number of a graph
Discussiones Mathematicae Graph Theory, 2007
Let G be a graph of order n with clique number ω(G), chromatic number χ(G) and independence number α(G). We show that χ(G) ≤ n+ω+1−α 2. Moreover, χ(G) ≤ n+ω−α 2 , if either ω + α = n + 1 and G is not a split graph or α+ω = n−1 and G contains no induced K ω+3 −C 5 .
An upper bound for the total chromatic number
Graphs and Combinatorics, 1990
The total chromatic number, Z"(G), of a graph G, is defined to be the minimum number ofcolours needed to colour the vertices and edges of a graph in such a way that no adjacent vertices, no adjacent edges and no incident vertex and edge are given the same colour. This paper shows that)('(G) _< z'(G) + 2x/~G), where z(G)is the vertex chromatic number and)((G)is the edge chromatic number of the graph.
A Bound on the Total Chromatic Number
COMBINATORICA, 1998
We prove that the total chromatic number of any graph with maximum degree is at most plus an absolute constant. In particular, we show that for su ciently large, the total chromatic number of such a graph is at most + 10 26. The proof is probabilistic.