Results 1  10
of
25
The timproper chromatic number of random graphs
, 2009
"... We consider the timproper chromatic number of the ErdősRényi random graph Gn,p. The timproper chromatic number χ t (G) is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most t. If t = 0, then this is the usua ..."
Abstract

Cited by 22 (13 self)
 Add to MetaCart
We consider the timproper chromatic number of the ErdősRényi random graph Gn,p. The timproper chromatic number χ t (G) is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most t. If t = 0, then this is the usual notion of proper colouring. When the edge probability p is constant, we provide a detailed description of the asymptotic behaviour of χ t (Gn,p) over the range of choices for the growth of t = t(n).
About a Brookstype theorem for improper colouring
, 2009
"... A graph is kimproperly ℓcolourable if its vertices can be partitioned into ℓ parts such that each part induces a subgraph of maximum degree at most k. A result of Lovász states that for any graph G, such a partition exists if ℓ ≥ ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
A graph is kimproperly ℓcolourable if its vertices can be partitioned into ℓ parts such that each part induces a subgraph of maximum degree at most k. A result of Lovász states that for any graph G, such a partition exists if ℓ ≥
Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k
, 2010
"... A graph G is (k, 0)colorable if its vertices can be partitioned into subsets V1 and V2 such that in G[V1] every vertex has degree at most k, while G[V2] is edgeless. For every integer k ≥ 1, we prove that every graph with the maximum average degree smaller than 3k+4 is (k,0)colorable. k+2 In parti ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
(Show Context)
A graph G is (k, 0)colorable if its vertices can be partitioned into subsets V1 and V2 such that in G[V1] every vertex has degree at most k, while G[V2] is edgeless. For every integer k ≥ 1, we prove that every graph with the maximum average degree smaller than 3k+4 is (k,0)colorable. k+2 In particular, it follows that every planar graph with girth at least 7 is (8,0)colorable. On the other hand, we construct planar graphs with girth 6 that are not (k, 0)colorable for arbitrarily large k.
Improper colouring of unit disk graphs
 In Proceedings of the 7th International Conference on Graph Theory, Electronic Notes in Discrete Mathematics
, 2005
"... We investigate the following problem proposed by Alcatel. A satellite sends information to receivers on earth, each of which is listening on a chosen frequency. Technically, it is impossible for the satellite to precisely focus its signal onto a receiver. Part of the signal will be spread in an area ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
(Show Context)
We investigate the following problem proposed by Alcatel. A satellite sends information to receivers on earth, each of which is listening on a chosen frequency. Technically, it is impossible for the satellite to precisely focus its signal onto a receiver. Part of the signal will be spread in an area around its destination and this creates noise for nearby receivers
A Note on List Improper Coloring Planar Graphs
 Appl. Math. Let
"... Communicated by R. L. Grahm AbstractA graph G is called (k, d)*choosable if, for every list assignment L satisfying [L(v)l = k for all v E V(G), there is an Lcoloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. In this note, we prove that every ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
Communicated by R. L. Grahm AbstractA graph G is called (k, d)*choosable if, for every list assignment L satisfying [L(v)l = k for all v E V(G), there is an Lcoloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. In this note, we prove that every planar graph without 4cycles and
EXTENSIONS OF A SIMPLE COMPETITIVE GRAPH COLORING ALGORITHM
, 2002
"... The (r, d)relaxed coloring game was recently introduced and studied by Chou, Wang, and Zhu. This game combines the original coloring game introduced by Bodlaender with the notion of defective coloring. Let G be a finite graph and let r and d be natural numbers. Two players, Alice and Bob, alternate ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
The (r, d)relaxed coloring game was recently introduced and studied by Chou, Wang, and Zhu. This game combines the original coloring game introduced by Bodlaender with the notion of defective coloring. Let G be a finite graph and let r and d be natural numbers. Two players, Alice and Bob, alternately color the uncolored vertices G with colors from a finite set X, with X  = r. At each step, for each α ∈ X, the subgraph induced by all vertices colored α must have maximum degree at most d, where d is called the defect. We say that Alice wins this game if all of the vertices of G are eventually colored; otherwise, Bob wins when there is an uncolored vertex which cannot be legally colored. In this dissertation, various classes of graphs are analyzed with respect to the (r, d)relaxed coloring game. The focus is on two goals. The first is to fix the defect and minimize the number of colors such that Alice has a winning strategy. The second is to find the least number of colors such that the defect can be bounded for a class of graphs. Most of the results will utilize and extend the activation techniques developed by Kierstead and Zhu over a series of articles. Results are proven for trees, outerplanar graphs, planar graphs, line
Good edgelabelling of graphs
 DISCRETE APPLIED MATHEMATICS
, 2011
"... A good edgelabelling of a graph G is a labelling of its edges such that, for any ordered pair of vertices (x,y), there do not exist two paths from x to y with increasing labels. This notion was introduced in [2] to solve wavelength assignment problems for specific categories of graphs. In this pape ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
A good edgelabelling of a graph G is a labelling of its edges such that, for any ordered pair of vertices (x,y), there do not exist two paths from x to y with increasing labels. This notion was introduced in [2] to solve wavelength assignment problems for specific categories of graphs. In this paper, we aim at characterizing the class of graphs that admit a good edgelabelling. First, we exhibit infinite families of graphs for which no such edgelabelling can be found. We then show that deciding if a graph G admits a good edgelabelling is NPcomplete, even if G is bipartite. Finally, we give large classes of graphs admitting a good edgelabelling: C3free outerplanar graphs, planar graphs of girth at least 6, {C3,K2,3}free subcubic graphs and {C3,K2,3}free ABCgraphs.
(k, 1)coloring of sparse graphs
 DISCRETE MATH
, 2009
"... A graph G is called (k,1)colorable, if the vertex set of G can be partitioned into subsets V1 and V2 such that the graph G[V1] induced by the vertices of V1 has maximum degree at most k and the graph G[V2] induced by the vertices of V2 has maximum degree at most 1. We prove that every graph with a ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
A graph G is called (k,1)colorable, if the vertex set of G can be partitioned into subsets V1 and V2 such that the graph G[V1] induced by the vertices of V1 has maximum degree at most k and the graph G[V2] induced by the vertices of V2 has maximum degree at most 1. We prove that every graph with a maximum average degree less than 10k+22 3k+9 admits a (k, 1)coloring, where k ≥ 2. In particular, every planar graph with girth at least 7 is (2, 1)colorable, while every planar graph with girth at least 6 is (5,1)colorable. On the other hand, for each k ≥ 2 we construct non(k, 1)colorable graphs whose maximum average degree is arbitrarily close to 14k 4k+1.