11,926 research outputs found
Planar graphs are 9/2-colorable
We show that every planar graph has a 2-fold 9-coloring. In particular,
this implies that has fractional chromatic number at most . This
is the first proof (independent of the 4 Color Theorem) that there exists a
constant such that every planar has fractional chromatic number at
most .Comment: 12 pages, 6 figures; following the suggestion of an editor, we split
the original version of this paper into two papers: one is the current
version of this paper, and the other is "Planar graphs have Independence
Ratio at least 3/13" (also available on the arXiv
The Clique Problem in Ray Intersection Graphs
Ray intersection graphs are intersection graphs of rays, or halflines, in the
plane. We show that any planar graph has an even subdivision whose complement
is a ray intersection graph. The construction can be done in polynomial time
and implies that finding a maximum clique in a segment intersection graph is
NP-hard. This solves a 21-year old open problem posed by Kratochv\'il and
Ne\v{s}et\v{r}il.Comment: 12 pages, 7 figure
Constructing dense graphs with sublinear Hadwiger number
Mader asked to explicitly construct dense graphs for which the size of the
largest clique minor is sublinear in the number of vertices. Such graphs exist
as a random graph almost surely has this property. This question and variants
were popularized by Thomason over several articles. We answer these questions
by showing how to explicitly construct such graphs using blow-ups of small
graphs with this property. This leads to the study of a fractional variant of
the clique minor number, which may be of independent interest.Comment: 10 page
Degree Sequence Index Strategy
We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by
which to bound graph invariants by certain indices in the ordered degree
sequence. As an illustration of the DSI strategy, we show how it can be used to
give new upper and lower bounds on the -independence and the -domination
numbers. These include, among other things, a double generalization of the
annihilation number, a recently introduced upper bound on the independence
number. Next, we use the DSI strategy in conjunction with planarity, to
generalize some results of Caro and Roddity about independence number in planar
graphs. Lastly, for claw-free and -free graphs, we use DSI to
generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester
Reoptimization of Some Maximum Weight Induced Hereditary Subgraph Problems
The reoptimization issue studied in this paper can be described as follows: given an instance I of some problem Π, an optimal solution OPT for Πin I and an instance I′ resulting from a local perturbation of I that consists of insertions or removals of a small number of data, we wish to use OPT in order to solve Πin I', either optimally or by guaranteeing an approximation ratio better than that guaranteed by an ex nihilo computation and with running time better than that needed for such a computation. We use this setting in order to study weighted versions of several representatives of a broad class of problems known in the literature as maximum induced hereditary subgraph problems. The main problems studied are max independent set, max k-colorable subgraph and max split subgraph under vertex insertions and deletion
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