15 research outputs found
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
The structure and the list 3-dynamic coloring of outer-1-planar graphs
An outer-1-planar graph is a graph admitting a drawing in the plane so that
all vertices appear in the outer region of the drawing and every edge crosses
at most one other edge. This paper establishes the local structure of
outer-1-planar graphs by proving that each outer-1-planar graph contains one of
the seventeen fixed configurations, and the list of those configurations is
minimal in the sense that for each fixed configuration there exist
outer-1-planar graphs containing this configuration that do not contain any of
another sixteen configurations. There are two interesting applications of this
structural theorem. First of all, we conclude that every (resp.maximal)
outer-1-planar graph of minimum degree at least 2 has an edge with the sum of
the degrees of its two end-vertices being at most 9 (resp.7), and this upper
bound is sharp. On the other hand, we show that the list 3-dynamic chromatic
number of every outer-1-planar graph is at most 6, and this upper bound is best
possible
Dynamic Chromatic Number of Bipartite Graphs
A dynamic coloring of a graph G is a proper vertex coloring such that for every vertex v Î 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 dynamic coloring with k colors, is called the dynamic chromatic number of G and denoted by c2(G). Montgomery conjectured that for every r-regular graph G, c2(G)-c(G) ≤ 2 . Finding an optimal upper bound for c2(G)-c(G) seems to be an intriguing problem. We show that there is a constant d such that every bipartite graph G with d(G) ³ d , has c2(G)-c(G) ≤ 2é(D(G))/(d(G))ù. It was shown that c2(G)-c(G) ≤ a' (G) +k* . Also, c2(G)-c(G) ≤ a(G) +k* . We prove that if G is a simple graph with d(G)>2, then c2(G)-c(G) ≤ (a' (G)+w(G) )/2 +k* . Among other results, we prove that for a given bipartite graph G=[X,Y], determining whether G has a dynamic 4-coloring l : V (G)®{a, b, c, d} such that a, b are used for part X and c, d are used for part Y is NP-complete