971 research outputs found
Defective Coloring on Classes of Perfect Graphs
In Defective Coloring we are given a graph and two integers ,
and are asked if we can -color so that the maximum
degree induced by any color class is at most . We show that this
natural generalization of Coloring is much harder on several basic graph
classes. In particular, we show that it is NP-hard on split graphs, even when
one of the two parameters , is set to the smallest possible
fixed value that does not trivialize the problem ( or ). Together with a simple treewidth-based DP algorithm this completely
determines the complexity of the problem also on chordal graphs. We then
consider the case of cographs and show that, somewhat surprisingly, Defective
Coloring turns out to be one of the few natural problems which are NP-hard on
this class. We complement this negative result by showing that Defective
Coloring is in P for cographs if either or is fixed; that
it is in P for trivially perfect graphs; and that it admits a sub-exponential
time algorithm for cographs when both and are unbounded
Parameterized Complexity of Equitable Coloring
A graph on vertices is equitably -colorable if it is -colorable and
every color is used either or times.
Such a problem appears to be considerably harder than vertex coloring, being
even for cographs and interval graphs.
In this work, we prove that it is for block
graphs and for disjoint union of split graphs when parameterized by the number
of colors; and for -free interval graphs
when parameterized by treewidth, number of colors and maximum degree,
generalizing a result by Fellows et al. (2014) through a much simpler
reduction.
Using a previous result due to Dominique de Werra (1985), we establish a
dichotomy for the complexity of equitable coloring of chordal graphs based on
the size of the largest induced star.
Finally, we show that \textsc{equitable coloring} is when
parameterized by the treewidth of the complement graph
Grid Representations and the Chromatic Number
A grid drawing of a graph maps vertices to grid points and edges to line
segments that avoid grid points representing other vertices. We show that there
is a number of grid points that some line segment of an arbitrary grid drawing
must intersect. This number is closely connected to the chromatic number.
Second, we study how many columns we need to draw a graph in the grid,
introducing some new \NP-complete problems. Finally, we show that any planar
graph has a planar grid drawing where every line segment contains exactly two
grid points. This result proves conjectures asked by David Flores-Pe\~naloza
and Francisco Javier Zaragoza Martinez.Comment: 22 pages, 8 figure
Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs
A path in an edge-colored graph is rainbow if no two edges of it are
colored the same. The graph is rainbow-connected if there is a rainbow path
between every pair of vertices. If there is a rainbow shortest path between
every pair of vertices, the graph is strongly rainbow-connected. The
minimum number of colors needed to make rainbow-connected is known as the
rainbow connection number of , and is denoted by . Similarly,
the minimum number of colors needed to make strongly rainbow-connected is
known as the strong rainbow connection number of , and is denoted by
. We prove that for every , deciding whether
is NP-complete for split graphs, which form a subclass
of chordal graphs. Furthermore, there exists no polynomial-time algorithm for
approximating the strong rainbow connection number of an -vertex split graph
with a factor of for any unless P = NP. We
then turn our attention to block graphs, which also form a subclass of chordal
graphs. We determine the strong rainbow connection number of block graphs, and
show it can be computed in linear time. Finally, we provide a polynomial-time
characterization of bridgeless block graphs with rainbow connection number at
most 4.Comment: 13 pages, 3 figure
Wide partitions, Latin tableaux, and Rota's basis conjecture
Say that mu is a ``subpartition'' of an integer partition lambda if the
multiset of parts of mu is a submultiset of the parts of lambda, and define an
integer partition lambda to be ``wide'' if for every subpartition mu of lambda,
mu >= mu' in dominance order (where mu' denotes the conjugate or transpose of
mu). Then Brian Taylor and the first author have conjectured that an integer
partition lambda is wide if and only if there exists a tableau of shape lambda
such that (1) for all i, the entries in the ith row of the tableau are
precisely the integers from 1 to lambda_i inclusive, and (2) for all j, the
entries in the jth column of the tableau are pairwise distinct. This conjecture
was originally motivated by Rota's basis conjecture and, if true, yields a new
class of integer multiflow problems that satisfy max-flow min-cut and
integrality. Wide partitions also yield a class of graphs that satisfy
``delta-conjugacy'' (in the sense of Greene and Kleitman), and the above
conjecture implies that these graphs furthermore have a completely saturated
stable set partition. We present several partial results, but the conjecture
remains very much open.Comment: Joined forces with Goemans and Vondrak---several new partial results;
28 pages, submitted to Adv. Appl. Mat
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