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On Some Cycles in Wenger Graphs
Let p be a prime, q be a power of p, and let F q be the field of q elements. For any
positive integer n, the Wenger graph W n (q) is defined as follows: it is a bipartite
graph with the vertex partitions being two copies of the (n + 1)-dimensional vector
space F n+1
, and two vertices p = (p(1), . . . , p(n + 1)), and l = [l(1), . . . , l(n + 1)]
q
being adjacent if p(i) + l(i) = p(1)l(1) i−1 , for all i = 2, 3, . . . , n + 1
An theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions
We introduce and develop a theory of limits for sequences of sparse graphs
based on graphons, which generalizes both the existing theory
of dense graph limits and its extension by Bollob\'as and Riordan to sparse
graphs without dense spots. In doing so, we replace the no dense spots
hypothesis with weaker assumptions, which allow us to analyze graphs with power
law degree distributions. This gives the first broadly applicable limit theory
for sparse graphs with unbounded average degrees. In this paper, we lay the
foundations of the theory of graphons, characterize convergence, and
develop corresponding random graph models, while we prove the equivalence of
several alternative metrics in a companion paper.Comment: 44 page
On clique-colouring of graphs with few P4's
Abstract
Let G=(V,E) be a graph with n vertices. A clique-colouring of a graph is a colouring of its vertices such that no maximal clique of size at least two is monocoloured. A k-clique-colouring is a clique-colouring that uses k colours. The clique-chromatic number of a graph G is the minimum k such that G has a k-clique-colouring.
In this paper we will use the primeval decomposition technique to find the clique-chromatic number and the clique-colouring of well known classes of graphs that in some local sense contain few P
4's. In particular we shall consider the classes of extended P
4-laden graphs, p-trees (graphs which contain exactly n−3 P
4's) and (q,q−3)-graphs, q≥7, such that no set of at most q vertices induces more that q−3 distincts P
4's. As corollary we shall derive the clique-chromatic number and the clique-colouring of the classes of cographs, P
4-reducible graphs, P
4-sparse graphs, extended P
4-reducible graphs, extended P
4-sparse graphs, P
4-extendible graphs, P
4-lite graphs, P
4-tidy graphs and P
4-laden graphs that are included in the class of extended P
4-laden graphs
Henneberg constructions and covers of cone-Laman graphs
We give Henneberg-type constructions for three families of sparse colored
graphs arising in the rigidity theory of periodic and other forced symmetric
frameworks. The proof method, which works with Laman-sparse finite covers of
colored graphs highlights the connection between these sparse colored families
and the well-studied matroidal (k, l)-sparse families.Comment: 14 pages, 2 figure
Continuous Yao Graphs
In this paper, we introduce a variation of the well-studied Yao graphs. Given
a set of points and an angle , we
define the continuous Yao graph with vertex set and angle
as follows. For each , we add an edge from to in
if there exists a cone with apex and aperture such
that is the closest point to inside this cone.
We study the spanning ratio of for different values of .
Using a new algebraic technique, we show that is a spanner when
. We believe that this technique may be of independent
interest. We also show that is not a spanner, and that
may be disconnected for .Comment: 7 pages, 7 figures. Presented at CCCG 201
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