961,563 research outputs found
Isometric embeddings of Johnson graphs in Grassmann graphs
Let be an -dimensional vector space () and let
be the Grassmannian formed by all -dimensional
subspaces of . The corresponding Grassmann graph will be denoted by
. We describe all isometric embeddings of Johnson graphs
, in , (Theorem 4). As a
consequence, we get the following: the image of every isometric embedding of
in is an apartment of if and
only if . Our second result (Theorem 5) is a classification of rigid
isometric embeddings of Johnson graphs in , .Comment: New version -- 14 pages accepted to Journal of Algebraic
Combinatoric
Near-colorings: non-colorable graphs and NP-completeness
A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned
into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of
V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus
on complexity aspects of such colorings when l=2,3. More precisely, we prove
that, for any fixed integers k,j,g with (k,j) distinct form (0,0) and g >= 3,
either every planar graph with girth at least g is (k,j)-colorable or it is
NP-complete to determine whether a planar graph with girth at least g is
(k,j)-colorable. Also, for any fixed integer k, it is NP-complete to determine
whether a planar graph that is either (0,0,0)-colorable or
non-(k,k,1)-colorable is (0,0,0)-colorable. Additionally, we exhibit
non-(3,1)-colorable planar graphs with girth 5 and non-(2,0)-colorable planar
graphs with girth 7
Non-singular circulant graphs and digraphs
We give necessary and sufficient conditions for a few classes of known
circulant graphs and/or digraphs to be singular. The above graph classes are
generalized to -digraphs for non-negative integers and , and
the digraph , with certain restrictions. We also obtain a
necessary and sufficient condition for the digraphs to be
singular. Some necessary conditions are given under which the
-digraphs are singular.Comment: 12 page
On Two problems of defective choosability
Given positive integers , and a non-negative integer , we say a
graph is -choosable if for every list assignment with
for each and ,
there exists an -coloring of such that each monochromatic subgraph has
maximum degree at most . In particular, -choosable means
-colorable, -choosable means -choosable and
-choosable means -defective -choosable. This paper proves
that there are 1-defective 3-choosable graphs that are not 4-choosable, and for
any positive integers , and non-negative integer , there
are -choosable graphs that are not -choosable.
These results answer questions asked by Wang and Xu [SIAM J. Discrete Math. 27,
4(2013), 2020-2037], and Kang [J. Graph Theory 73, 3(2013), 342-353],
respectively. Our construction of -choosable but not -choosable graphs generalizes the construction of Kr\'{a}l' and Sgall
in [J. Graph Theory 49, 3(2005), 177-186] for the case .Comment: 12 pages, 4 figure
Spectrum of Markov generators on sparse random graphs
Correction in Proposition 4.3. Final version.International audienceWe investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph. This is the non-Hermitian random nxn matrix L defined by L(j,k)=X(j,k) if kj and L(j,j)=-sum(L(j,k),kj), where X(j,k) are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n tends to infinity of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erdős-Rényi graph where each edge is present independently with probability p(n)->0 as long as np(n) >> (log(n))^6. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L
Approximating subset -connectivity problems
A subset of terminals is -connected to a root in a
directed/undirected graph if has internally-disjoint -paths for
every ; is -connected in if is -connected to every
. We consider the {\sf Subset -Connectivity Augmentation} problem:
given a graph with edge/node-costs, node subset , and
a subgraph of such that is -connected in , find a
minimum-cost augmenting edge-set such that is
-connected in . The problem admits trivial ratio .
We consider the case and prove that for directed/undirected graphs and
edge/node-costs, a -approximation for {\sf Rooted Subset -Connectivity
Augmentation} implies the following ratios for {\sf Subset -Connectivity
Augmentation}: (i) ; (ii) , where
b=1 for undirected graphs and b=2 for directed graphs, and is the th
harmonic number. The best known values of on undirected graphs are
for edge-costs and for
node-costs; for directed graphs for both versions. Our results imply
that unless , {\sf Subset -Connectivity Augmentation} admits
the same ratios as the best known ones for the rooted version. This improves
the ratios in \cite{N-focs,L}
Self-Similar Corrections to the Ergodic Theorem for the Pascal-Adic Transformation
Let T be the Pascal-adic transformation. For any measurable function g, we
consider the corrections to the ergodic theorem sum_{k=0}^{j-1} g(T^k x) - j/l
sum_{k=0}^{l-1} g(T^k x). When seen as graphs of functions defined on
{0,...,l-1}, we show for a suitable class of functions g that these quantities,
once properly renormalized, converge to (part of) the graph of a self-affine
function. The latter only depends on the ergodic component of x, and is a
deformation of the so-called Blancmange function. We also briefly describe the
links with a series of works on Conway recursive recursive sequenc
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