8 research outputs found
A note on connected greedy edge colouring
Following a given ordering of the edges of a graph , the greedy edge
colouring procedure assigns to each edge the smallest available colour. The
minimum number of colours thus involved is the chromatic index , and
the maximum is the so-called Grundy chromatic index. Here, we are interested in
the restricted case where the ordering of the edges builds the graph in a
connected fashion. Let be the minimum number of colours involved
following such an ordering. We show that it is NP-hard to determine whether
. We prove that if is bipartite,
and that if is subcubic.Comment: Comments welcome, 12 page
Complexity of Grundy coloring and its variants
The Grundy number of a graph is the maximum number of colors used by the
greedy coloring algorithm over all vertex orderings. In this paper, we study
the computational complexity of GRUNDY COLORING, the problem of determining
whether a given graph has Grundy number at least . We also study the
variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper)
and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring
algorithm, the subgraph induced by the colored vertices must be connected).
We show that GRUNDY COLORING can be solved in time and WEAK
GRUNDY COLORING in time on graphs of order . While GRUNDY
COLORING and WEAK GRUNDY COLORING are known to be solvable in time
for graphs of treewidth (where is the number of
colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot
be solved in time . We also describe an
algorithm for WEAK GRUNDY COLORING, which is therefore
\fpt for the parameter . Moreover, under the ETH, we prove that such a
running time is essentially optimal (this lower bound also holds for GRUNDY
COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we
show that this is the case for graphs belonging to a number of standard graph
classes including chordal graphs, claw-free graphs, and graphs excluding a
fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY
COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with
the two other problems, we show that CONNECTED GRUNDY COLORING is
\np-complete already for colors.Comment: 24 pages, 7 figures. This version contains some new results and
improvements. A short paper based on version v2 appeared in COCOON'1
A new vertex coloring heuristic and corresponding chromatic number
One method to obtain a proper vertex coloring of graphs using a reasonable
number of colors is to start from any arbitrary proper coloring and then repeat
some local re-coloring techniques to reduce the number of color classes. The
Grundy (First-Fit) coloring and color-dominating colorings of graphs are two
well-known such techniques. The color-dominating colorings are also known and
commonly referred as {\rm b}-colorings. But these two topics have been studied
separately in graph theory. We introduce a new coloring procedure which
combines the strategies of these two techniques and satisfies an additional
property. We first prove that the vertices of every graph can be
effectively colored using color classes say such that
for any two colors and with , any vertex of color
is adjacent to a vertex of color , there exists a set of vertices of such that for any and is adjacent to for each with
, and for each and with , the vertex
has a neighbor in . This provides a new vertex coloring heuristic which
improves both Grundy and color-dominating colorings. Denote by the
maximum number of colors used in any proper vertex coloring satisfying the
above properties. The quantifies the worst-case behavior of the
heuristic. We prove the existence of such that but for each .
For each positive integer we construct a family of finitely many colored
graphs satisfying the property that if for a
graph then contains an element from as a colored
subgraph. This provides an algorithmic method for proving numeric upper bounds
for
Connected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly
8 pages, 3 figuresThe Grundy number of a graph is the maximum number of colours used by the ``First-Fit'' greedy colouring algorithm over all vertex orderings. Given a vertex ordering , the ``First-Fit'' greedy colouring algorithm colours the vertices in the order of by assigning to each vertex the smallest colour unused in its neighbourhood. By restricting this procedure to vertex orderings that are connected, we obtain {\em connected greedy colourings}. For some graphs, all connected greedy colourings use exactly colours; they are called {\em good graphs}. On the opposite, some graphs do not admit any connected greedy colouring using only colours; they are called {\em ugly graphs}. We show that no perfect graph is ugly. We also give simple proofs of this fact for subclasses of perfect graphs (block graphs, comparability graphs), and show that no -minor free graph is ugly
Connected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly
8 pages, 3 figuresThe Grundy number of a graph is the maximum number of colours used by the ``First-Fit'' greedy colouring algorithm over all vertex orderings. Given a vertex ordering , the ``First-Fit'' greedy colouring algorithm colours the vertices in the order of by assigning to each vertex the smallest colour unused in its neighbourhood. By restricting this procedure to vertex orderings that are connected, we obtain {\em connected greedy colourings}. For some graphs, all connected greedy colourings use exactly colours; they are called {\em good graphs}. On the opposite, some graphs do not admit any connected greedy colouring using only colours; they are called {\em ugly graphs}. We show that no perfect graph is ugly. We also give simple proofs of this fact for subclasses of perfect graphs (block graphs, comparability graphs), and show that no -minor free graph is ugly