25,045 research outputs found
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
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
Unavoidable induced subgraphs in large graphs with no homogeneous sets
A homogeneous set of an -vertex graph is a set of vertices () such that every vertex not in is either complete or
anticomplete to . A graph is called prime if it has no homogeneous set. A
chain of length is a sequence of vertices such that for every vertex
in the sequence except the first one, its immediate predecessor is its unique
neighbor or its unique non-neighbor among all of its predecessors. We prove
that for all , there exists such that every prime graph with at least
vertices contains one of the following graphs or their complements as an
induced subgraph: (1) the graph obtained from by subdividing every
edge once, (2) the line graph of , (3) the line graph of the graph in
(1), (4) the half-graph of height , (5) a prime graph induced by a chain of
length , (6) two particular graphs obtained from the half-graph of height
by making one side a clique and adding one vertex.Comment: 13 pages, 3 figure
A sharp threshold for random graphs with a monochromatic triangle in every edge coloring
Let be the set of all finite graphs with the Ramsey property that
every coloring of the edges of by two colors yields a monochromatic
triangle. In this paper we establish a sharp threshold for random graphs with
this property. Let be the random graph on vertices with edge
probability . We prove that there exists a function with
, as tends to infinity
Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat
c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is
of independent interest is a generalization of Szemer\'edi's Regularity Lemma
to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.
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