10,641 research outputs found
Planar Induced Subgraphs of Sparse Graphs
We show that every graph has an induced pseudoforest of at least
vertices, an induced partial 2-tree of at least vertices, and an
induced planar subgraph of at least vertices. These results are
constructive, implying linear-time algorithms to find the respective induced
subgraphs. We also show that the size of the largest -minor-free graph in
a given graph can sometimes be at most .Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph
Algorithms and Application
Sudden emergence of q-regular subgraphs in random graphs
We investigate the computationally hard problem whether a random graph of
finite average vertex degree has an extensively large -regular subgraph,
i.e., a subgraph with all vertices having degree equal to . We reformulate
this problem as a constraint-satisfaction problem, and solve it using the
cavity method of statistical physics at zero temperature. For , we find
that the first large -regular subgraphs appear discontinuously at an average
vertex degree c_\reg{3} \simeq 3.3546 and contain immediately about 24% of
all vertices in the graph. This transition is extremely close to (but different
from) the well-known 3-core percolation point c_\cor{3} \simeq 3.3509. For
, the -regular subgraph percolation threshold is found to coincide with
that of the -core.Comment: 7 pages, 5 figure
On the threshold for k-regular subgraphs of random graphs
The -core of a graph is the largest subgraph of minimum degree at least
. We show that for sufficiently large, the -core of a random
graph \G(n,p) asymptotically almost surely has a spanning -regular
subgraph. Thus the threshold for the appearance of a -regular subgraph of a
random graph is at most the threshold for the -core. In particular, this
pins down the point of appearance of a -regular subgraph in \G(n,p) to a
window for of width roughly for large and moderately large
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
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