173 research outputs found
On coloring parameters of triangle-free planar -graphs
An -graph is a graph with types of arcs and types of edges. A
homomorphism of an -graph to another -graph is a vertex
mapping that preserves the adjacencies along with their types and directions.
The order of a smallest (with respect to the number of vertices) such is
the -chromatic number of .Moreover, an -relative clique of
an -graph is a vertex subset of for which no two distinct
vertices of get identified under any homomorphism of . The
-relative clique number of , denoted by , is the
maximum such that is an -relative clique of . In practice,
-relative cliques are often used for establishing lower bounds of
-chromatic number of graph families.
Generalizing an open problem posed by Sopena [Discrete Mathematics 2016] in
his latest survey on oriented coloring, Chakroborty, Das, Nandi, Roy and Sen
[Discrete Applied Mathematics 2022] conjectured that for any triangle-free planar -graph and that this
bound is tight for all .In this article, we positively settle
this conjecture by improving the previous upper bound of to , and by
finding examples of triangle-free planar graphs that achieve this bound. As a
consequence of the tightness proof, we also establish a new lower bound of for the -chromatic number for the family of triangle-free
planar graphs.Comment: 22 Pages, 5 figure
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem
We consider arrangements of axis-aligned rectangles in the plane. A geometric
arrangement specifies the coordinates of all rectangles, while a combinatorial
arrangement specifies only the respective intersection type in which each pair
of rectangles intersects. First, we investigate combinatorial contact
arrangements, i.e., arrangements of interior-disjoint rectangles, with a
triangle-free intersection graph. We show that such rectangle arrangements are
in bijection with the 4-orientations of an underlying planar multigraph and
prove that there is a corresponding geometric rectangle contact arrangement.
Moreover, we prove that every triangle-free planar graph is the contact graph
of such an arrangement. Secondly, we introduce the question whether a given
rectangle arrangement has a combinatorially equivalent square arrangement. In
addition to some necessary conditions and counterexamples, we show that
rectangle arrangements pierced by a horizontal line are squarable under certain
sufficient conditions.Comment: 15 pages, 13 figures, extended version of a paper to appear at the
International Symposium on Graph Drawing and Network Visualization (GD) 201
{\Gamma}-species, quotients, and graph enumeration
The theory of {\Gamma}-species is developed to allow species-theoretic study
of quotient structures in a categorically rigorous fashion. This new approach
is then applied to two graph-enumeration problems which were previously
unsolved in the unlabeled case-bipartite blocks and general k-trees.Comment: 84 pages, 10 figures, dissertatio
The complexity of deciding whether a graph admits an orientation with fixed weak diameter
International audienceAn oriented graph is said weak (resp. strong) if, for every pair of vertices of , there are directed paths joining and in either direction (resp. both directions). In case, for every pair of vertices, some of these directed paths have length at most , we call -weak (resp. -strong). We consider several problems asking whether an undirected graph admits orientations satisfying some connectivity and distance properties. As a main result, we show that deciding whether admits a -weak orientation is NP-complete for every . This notably implies the NP-completeness of several problems asking whether is an extremal graph (in terms of needed colours) for some vertex-colouring problems
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