1,592 research outputs found
Problems in extremal graph theory
We consider a variety of problems in extremal graph and set theory.
The {\em chromatic number} of , , is the smallest integer
such that is -colorable.
The {\it square} of , written , is the supergraph of in which also
vertices within distance 2 of each other in are adjacent.
A graph is a {\it minor} of if
can be obtained from a subgraph of by contracting edges.
We show that the upper bound for
conjectured by Wegner (1977) for planar graphs
holds when is a -minor-free graph.
We also show that is equal to the bound
only when contains a complete graph of that order.
One of the central problems of extremal hypergraph theory is
finding the maximum number of edges in a hypergraph
that does not contain a specific forbidden structure.
We consider as a forbidden structure a fixed number of members
that have empty common intersection
as well as small union.
We obtain a sharp upper bound on the size of uniform hypergraphs
that do not contain this structure,
when the number of vertices is sufficiently large.
Our result is strong enough to imply the same sharp upper bound
for several other interesting forbidden structures
such as the so-called strong simplices and clusters.
The {\em -dimensional hypercube}, ,
is the graph whose vertex set is and
whose edge set consists of the vertex pairs
differing in exactly one coordinate.
The generalized Tur\'an problem asks for the maximum number
of edges in a subgraph of a graph that does not contain
a forbidden subgraph .
We consider the Tur\'an problem where is and
is a cycle of length with .
Confirming a conjecture of Erd{\H o}s (1984),
we show that the ratio of the size of such a subgraph of
over the number of edges of is ,
i.e. in the limit this ratio approaches 0
as approaches infinity
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
Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)
We survey work on coloring, list coloring, and painting squares of graphs; in
particular, we consider strong edge-coloring. We focus primarily on planar
graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography,
comments are welcome, published as a Dynamic Survey in Electronic Journal of
Combinatoric
Coloring the Square of Planar Graphs Without 4-Cycles or 5-Cycles
The famous Four Color Theorem states that any planar graph can be properly colored using at most four colors. However, if we want to properly color the square of a planar graph (or alternatively, color the graph using distinct colors on vertices at distance up to two from each other), we will always require at least \Delta + 1 colors, where \Delta is the maximum degree in the graph. For all \Delta, Wegner constructed planar graphs (even without 3-cycles) that require about \frac{3}{2} \Delta colors for such a coloring.
To prove a stronger upper bound, we consider only planar graphs that contain no 4-cycles and no 5-cycles (but which may contain 3-cycles). Zhu, Lu, Wang, and Chen showed that for a graph G in this class with \Delta \ge 9, we can color G^2 using no more than \Delta + 5 colors. In this thesis we improve this result, showing that for a planar graph G with maximum degree \Delta \ge 32 having no 4-cycles and no 5-cycles, at most \Delta + 3 colors are needed to properly color G^2. Our approach uses the discharging method, and the result extends to list-coloring and other related coloring concepts as well
Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult
We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem
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