347 research outputs found
Induced subdivisions and bounded expansion
We prove that for every graph H and for every integer s, the class of graphs
that do not contain K_s, K_{s,s}, or any subdivision of H as an induced
subgraph has bounded expansion; this strengthens a result of Kuhn and Osthus.
The argument also gives another characterization of graph classes with bounded
expansion and of nowhere-dense graph classes.Comment: 8 pages, no figure
Thin graph classes and polynomial-time approximation schemes
Baker devised a powerful technique to obtain approximation schemes for
various problems restricted to planar graphs. Her technique can be directly
extended to various other graph classes, among the most general ones the graphs
avoiding a fixed apex graph as a minor. Further generalizations (e.g., to all
proper minor closed graph classes) are known, but they use a combination of
techniques and usually focus on somewhat restricted classes of problems. We
present a new type of graph decompositions (thin systems of overlays)
generalizing Baker's technique and leading to straightforward polynomial-time
approximation schemes. We also show that many graph classes (all proper
minor-closed classes, and all subgraph-closed classes with bounded maximum
degree and strongly sublinear separators) admit such decompositions.Comment: 30 pages, no figure
A simplified discharging proof of Gr\"otzsch theorem
In this note, we combine ideas of several previous proofs in order to obtain
a quite short proof of Gr\"otzsch theorem.Comment: 6 pages, 0 figure
Baker game and polynomial-time approximation schemes
Baker devised a technique to obtain approximation schemes for many
optimization problems restricted to planar graphs; her technique was later
extended to more general graph classes. In particular, using the Baker's
technique and the minor structure theorem, Dawar et al. gave Polynomial-Time
Approximation Schemes (PTAS) for all monotone optimization problems expressible
in the first-order logic when restricted to a proper minor-closed class of
graphs. We define a Baker game formalizing the notion of repeated application
of Baker's technique interspersed with vertex removal, prove that monotone
optimization problems expressible in the first-order logic admit PTAS when
restricted to graph classes in which the Baker game can be won in a constant
number of rounds, and prove without use of the minor structure theorem that all
proper minor-closed classes of graphs have this property.Comment: 27 pages, no figure
Planar graphs without cycles of length 4 or 5 are (11:3)-colorable
A graph G is (a:b)-colorable if there exists an assignment of b-element
subsets of {1,...,a} to vertices of G such that sets assigned to adjacent
vertices are disjoint. We show that every planar graph without cycles of length
4 or 5 is (11:3)-colorable, a weakening of recently disproved Steinberg's
conjecture. In particular, each such graph with n vertices has an independent
set of size at least 3n/11.Comment: 23 pages, 5 figures; incorporated reviewer remark
Fine structure of 4-critical triangle-free graphs III. General surfaces
Dvo\v{r}\'ak, Kr\'al' and Thomas gave a description of the structure of
triangle-free graphs on surfaces with respect to 3-coloring. Their description
however contains two substructures (both related to graphs embedded in plane
with two precolored cycles) whose coloring properties are not entirely
determined. In this paper, we fill these gaps.Comment: 15 pages, 1 figure; corrections from the review process. arXiv admin
note: text overlap with arXiv:1509.0101
4-critical graphs on surfaces without contractible (<=4)-cycles
We show that if G is a 4-critical graph embedded in a fixed surface
so that every contractible cycle has length at least 5, then G can be expressed
as , where and are bounded
by a constant (depending linearly on the genus of ) and
are graphs (of unbounded size) whose structure we describe exactly. The proof
is computer-assisted - we use computer to enumerate all plane 4-critical graphs
of girth 5 with a precolored cycle of length at most 16, that are used in the
basic case of the inductive proof of the statement.Comment: 52 pages, 19 figure
(3a:a)-list-colorability of embedded graphs of girth at least five
A graph G is list (b:a)-colorable if for every assignment of lists of size b
to vertices of G, there exists a choice of an a-element subset of the list at
each vertex such that the subsets chosen at adjacent vertices are disjoint. We
prove that for every positive integer a, the family of minimal obstructions of
girth at least five to list (3a:a)-colorability is strongly hyperbolic, in the
sense of the hyperbolicity theory developed by Postle and Thomas. This has a
number of consequences, e.g., that if a graph of girth at least five and Euler
genus g is not list (3a:a)-colorable, then G contains a subgraph with O(g)
vertices which is not list (3a:a) colorable.Comment: 32 pages, no figures; updated for reviewer comments, added a more
detailed hyperbolicity argument as an appendi
Fine structure of 4-critical triangle-free graphs II. Planar triangle-free graphs with two precolored 4-cycles
We study 3-coloring properties of triangle-free planar graphs with two
precolored 4-cycles and that are far apart. We prove that either
every precoloring of extends to a 3-coloring of , or
contains one of two special substructures which uniquely determine which
3-colorings of extend. As a corollary, we prove that there exists
a constant such that if is a planar triangle-free graph and
consists of vertices at pairwise distances at least , then
every precoloring of extends to a 3-coloring of . This gives a positive
answer to a conjecture of Dvo\v{r}\'ak, Kr\'al' and Thomas, and implies an
exponential lower bound on the number of 3-colorings of triangle-free planar
graphs of bounded maximum degree.Comment: 12 pages, 2 figure
Fractional coloring of planar graphs of girth five
A graph G is (a:b)-colorable if there exists an assignment of b-element
subsets of {1,...,a} to vertices of G such that sets assigned to adjacent
vertices are disjoint. We first show that for every triangle-free planar graph
G and a vertex x of G, the graph G has a set coloring phi by subsets of
{1,...,6} such that |phi(v)|>=2 for each vertex v of G and |phi(x)|=3. As a
corollary, every triangle-free planar graph on n vertices is
(6n:2n+1)-colorable. We further use this result to prove that for every Delta,
there exists a constant M_Delta such that every planar graph G of girth at
least five and maximum degree Delta is (6M_Delta:2M_Delta+1)-colorable.
Consequently, planar graphs of girth at least five with bounded maximum degree
Delta have fractional chromatic number at most 3-3/(2M_Delta+1).Comment: 19 pages, 3 figure
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