88 research outputs found
Cycle factors and renewal theory
For which values of does a uniformly chosen -regular graph on
vertices typically contain vertex-disjoint -cycles (a -cycle
factor)? To date, this has been answered for and for ; the
former, the Hamiltonicity problem, was finally answered in the affirmative by
Robinson and Wormald in 1992, while the answer in the latter case is negative
since with high probability most vertices do not lie on -cycles.
Here we settle the problem completely: the threshold for a -cycle factor
in as above is with . Precisely, we prove a 2-point concentration result: if divides then contains a -cycle factor
w.h.p., whereas if then w.h.p. it
does not. As a byproduct, we confirm the "Comb Conjecture," an old problem
concerning the embedding of certain spanning trees in the random graph
.
The proof follows the small subgraph conditioning framework, but the
associated second moment analysis here is far more delicate than in any earlier
use of this method and involves several novel features, among them a sharp
estimate for tail probabilities in renewal processes without replacement which
may be of independent interest.Comment: 45 page
Pancyclicity of 4-connected {Claw, Generalized Bull}-free Graphs
A graph G is pancyclic if it contains cycles of each length â, 3 †â †|V (G)|. The generalized bull B(i, j) is obtained by associating one endpoint of each of the paths P i+1 and P j+1 with distinct vertices of a triangle. Gould, Luczak and Pfende
Proof of Koml\'os's conjecture on Hamiltonian subsets
Koml\'os conjectured in 1981 that among all graphs with minimum degree at
least , the complete graph minimises the number of Hamiltonian
subsets, where a subset of vertices is Hamiltonian if it contains a spanning
cycle. We prove this conjecture when is sufficiently large. In fact we
prove a stronger result: for large , any graph with average degree at
least contains almost twice as many Hamiltonian subsets as ,
unless is isomorphic to or a certain other graph which we
specify.Comment: 33 pages, to appear in Proceedings of the London Mathematical Societ
Ramsey properties of randomly perturbed graphs: cliques and cycles
Given graphs , a graph is -Ramsey if for every
colouring of the edges of with red and blue, there is a red copy of
or a blue copy of . In this paper we investigate Ramsey questions in the
setting of randomly perturbed graphs: this is a random graph model introduced
by Bohman, Frieze and Martin in which one starts with a dense graph and then
adds a given number of random edges to it. The study of Ramsey properties of
randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali in
2006; they determined how many random edges must be added to a dense graph to
ensure the resulting graph is with high probability -Ramsey (for
). They also raised the question of generalising this result to pairs
of graphs other than . We make significant progress on this
question, giving a precise solution in the case when and
where . Although we again show that one requires polynomially fewer
edges than in the purely random graph, our result shows that the problem in
this case is quite different to the -Ramsey question. Moreover, we
give bounds for the corresponding -Ramsey question; together with a
construction of Powierski this resolves the -Ramsey problem.
We also give a precise solution to the analogous question in the case when
both and are cycles. Additionally we consider the
corresponding multicolour problem. Our final result gives another
generalisation of the Krivelevich, Sudakov and Tetali result. Specifically, we
determine how many random edges must be added to a dense graph to ensure the
resulting graph is with high probability -Ramsey (for odd
and ).Comment: 24 pages + 12-page appendix; v2: cited independent work of Emil
Powierski, stated results for cliques in graphs of low positive density
separately (Theorem 1.6) for clarity; v3: author accepted manuscript, to
appear in CP
A General Purpose Algorithm for Counting Simple Cycles and Simple Paths of Any Length
International audienceWe describe a general purpose algorithm for counting simple cycles and simple paths of any length on a (weighted di)graph on N vertices and M edges, achieving a time complexity of O N + M + Ï + â |S |. In this expression, |S | is the number of (weakly) connected induced subgraphs of G on at most vertices, â is the maximum degree of any vertex and Ï is the exponent of matrix multiplication. We compare the algorithm complexity both theoretically and experimentally with most of the existing algorithms for the same task. These comparisons show that the algorithm described here is the best general purpose algorithm for the class of graphs where (Ïâ1 â â1 +1)|S | †|Cycle |, with |Cycle | the total number of simple cycles of length at most , including backtracks and self-loops. On ErdĆs-RĂ©nyi random graphs, we find empirically that this happens when the edge probability is larger than circa 4/N. In addition, we show that some real-world networks also belong to this class. Finally, the algorithm permits the enumeration of simple cycles and simple paths on networks where vertices are labeled from an alphabet on n letters with a time complexity of O N + M + n Ï + â |S |. A Matlab implementation of the algorithm proposed here is available for download
Pancyclicity of highly connected graphs
A well-known result due to Chvat\'al and Erd\H{o}s (1972) asserts that, if a
graph satisfies , where is the
vertex-connectivity of , then has a Hamilton cycle. We prove a similar
result implying that a graph is pancyclic, namely it contains cycles of all
lengths between and : if is large and ,
then is pancyclic. This confirms a conjecture of Jackson and Ordaz (1990)
for large graphs, and improves upon a very recent result of Dragani\'c,
Munh\'a-Correia, and Sudakov.Comment: 31 pages, 11 figure
Mean quantum percolation
We study the spectrum of adjacency matrices of random graphs. We develop two techniques to lower bound the mass of the continuous part of the spectral measure or the density of states. As an application, we prove that the spectral measure of bond percolation in the twodimensional lattice contains a non-trivial continuous part in the supercritical regime. The same result holds for the limiting spectral measure of a supercritical Erd'os-Rényi graph and for the spectral measure of a unimodular random tree with at least two ends. We give examples of random graphs with purely continuous spectrum. © European Mathematical Society 2017
On prisms, M\"obius ladders and the cycle space of dense graphs
For a graph X, let f_0(X) denote its number of vertices, d(X) its minimum
degree and Z_1(X;Z/2) its cycle space in the standard graph-theoretical sense
(i.e. 1-dimensional cycle group in the sense of simplicial homology theory with
Z/2-coefficients). Call a graph Hamilton-generated if and only if the set of
all Hamilton circuits is a Z/2-generating system for Z_1(X;Z/2). The main
purpose of this paper is to prove the following: for every s > 0 there exists
n_0 such that for every graph X with f_0(X) >= n_0 vertices, (1) if d(X) >=
(1/2 + s) f_0(X) and f_0(X) is odd, then X is Hamilton-generated, (2) if d(X)
>= (1/2 + s) f_0(X) and f_0(X) is even, then the set of all Hamilton circuits
of X generates a codimension-one subspace of Z_1(X;Z/2), and the set of all
circuits of X having length either f_0(X)-1 or f_0(X) generates all of
Z_1(X;Z/2), (3) if d(X) >= (1/4 + s) f_0(X) and X is square bipartite, then X
is Hamilton-generated. All these degree-conditions are essentially
best-possible. The implications in (1) and (2) give an asymptotic affirmative
answer to a special case of an open conjecture which according to [European J.
Combin. 4 (1983), no. 3, p. 246] originates with A. Bondy.Comment: 33 pages; 5 figure
Combinatorics, Probability and Computing
One of the exciting phenomena in mathematics in recent years has been the widespread and surprisingly eïŹective use of probabilistic methods in diverse areas. The probabilistic point of view has turned out to b
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