545 research outputs found
Cliques and extended triangles : A necessary condition for planar clique graphs
By generalizing the idea of extended triangle of a graph, we succeed in obtaining a common framework for the result of Roberts and Spencer about clique graphs and the one of Szwarcfiter about Helly graphs. We characterize Helly and 3-Helly planar graphs using extended triangles. We prove that if a planar graph G is a clique graph, then every extended triangle of G must be a clique graph. Finally, we show the extended triangles of a planar graph which are clique graphs. Any one of the obtained characterizations are tested in O(n2) time.Facultad de Ciencias Exacta
Graphs with Plane Outside-Obstacle Representations
An \emph{obstacle representation} of a graph consists of a set of polygonal
obstacles and a distinct point for each vertex such that two points see each
other if and only if the corresponding vertices are adjacent. Obstacle
representations are a recent generalization of classical polygon--vertex
visibility graphs, for which the characterization and recognition problems are
long-standing open questions.
In this paper, we study \emph{plane outside-obstacle representations}, where
all obstacles lie in the unbounded face of the representation and no two
visibility segments cross. We give a combinatorial characterization of the
biconnected graphs that admit such a representation. Based on this
characterization, we present a simple linear-time recognition algorithm for
these graphs. As a side result, we show that the plane vertex--polygon
visibility graphs are exactly the maximal outerplanar graphs and that every
chordal outerplanar graph has an outside-obstacle representation.Comment: 12 pages, 7 figure
Decompositions of Triangle-Dense Graphs
High triangle density -- the graph property stating that a constant fraction
of two-hop paths belong to a triangle -- is a common signature of social
networks. This paper studies triangle-dense graphs from a structural
perspective. We prove constructively that significant portions of a
triangle-dense graph are contained in a disjoint union of dense, radius 2
subgraphs. This result quantifies the extent to which triangle-dense graphs
resemble unions of cliques. We also show that our algorithm recovers planted
clusterings in approximation-stable k-median instances.Comment: 20 pages. Version 1->2: Minor edits. 2->3: Strengthened {\S}3.5,
removed appendi
Ramsey numbers of cubes versus cliques
The cube graph Q_n is the skeleton of the n-dimensional cube. It is an
n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N
such that every graph of order N contains the cube graph Q_n or an independent
set of order s. Burr and Erdos in 1983 asked whether the simple lower bound
r(Q_n, K_s) >= (s-1)(2^n - 1)+1 is tight for s fixed and n sufficiently large.
We make progress on this problem, obtaining the first upper bound which is
within a constant factor of the lower bound.Comment: 26 page
Remarks on Schur's conjecture
Let P be a set of n>d points in Rd for d≥2. It was conjectured by Zvi Schur that the maximum number of (d-1)-dimensional regular simplices of edge length diam(P), whose every vertex belongs to P, is n. We prove this statement under the condition that any two of the simplices share at least d-2 vertices. It is left as an open question to decide whether this condition is always satisfied. We also establish upper bounds on the number of all 2- and 3-dimensional simplices induced by a set P⊂R3 of n points which satisfy the condition that the lengths of their sides belong to the set of k largest distances determined by P
Distributed coloring in sparse graphs with fewer colors
This paper is concerned with efficiently coloring sparse graphs in the
distributed setting with as few colors as possible. According to the celebrated
Four Color Theorem, planar graphs can be colored with at most 4 colors, and the
proof gives a (sequential) quadratic algorithm finding such a coloring. A
natural problem is to improve this complexity in the distributed setting. Using
the fact that planar graphs contain linearly many vertices of degree at most 6,
Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm
coloring -vertex planar graphs with 7 colors in rounds. Here, we
show how to color planar graphs with 6 colors in \mbox{polylog}(n) rounds.
Our algorithm indeed works more generally in the list-coloring setting and for
sparse graphs (for such graphs we improve by at least one the number of colors
resulting from an efficient algorithm of Barenboim and Elkin, at the expense of
a slightly worst complexity). Our bounds on the number of colors turn out to be
quite sharp in general. Among other results, we show that no distributed
algorithm can color every -vertex planar graph with 4 colors in
rounds.Comment: 16 pages, 4 figures - An extended abstract of this work was presented
at PODC'18 (ACM Symposium on Principles of Distributed Computing
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