12 research outputs found
Hamilton cycles in highly connected and expanding graphs
In this paper we prove a sufficient condition for the existence of a Hamilton
cycle, which is applicable to a wide variety of graphs, including relatively
sparse graphs. In contrast to previous criteria, ours is based on only two
properties: one requiring expansion of ``small'' sets, the other ensuring the
existence of an edge between any two disjoint ``large'' sets. We also discuss
applications in positional games, random graphs and extremal graph theory.Comment: 19 page
On covering expander graphs by Hamilton cycles
The problem of packing Hamilton cycles in random and pseudorandom graphs has
been studied extensively. In this paper, we look at the dual question of
covering all edges of a graph by Hamilton cycles and prove that if a graph with
maximum degree satisfies some basic expansion properties and contains
a family of edge disjoint Hamilton cycles, then there also
exists a covering of its edges by Hamilton cycles. This
implies that for every and every there exists
a covering of all edges of by Hamilton cycles
asymptotically almost surely, which is nearly optimal.Comment: 19 pages. arXiv admin note: some text overlap with arXiv:some
math/061275
Hamilton cycles in highly connected and expanding graphs
In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on two properties only: one requiring expansion of "small” sets, the other ensuring the existence of an edge between any two disjoint "large” sets. We also discuss applications in positional games, random graphs and extremal graph theor
Expanders Are Universal for the Class of All Spanning Trees
Given a class of graphs F, we say that a graph G is universal for F, or
F-universal, if every H in F is contained in G as a subgraph. The construction
of sparse universal graphs for various families F has received a considerable
amount of attention. One is particularly interested in tight F-universal
graphs, i.e., graphs whose number of vertices is equal to the largest number of
vertices in a graph from F. Arguably, the most studied case is that when F is
some class of trees.
Given integers n and \Delta, we denote by T(n,\Delta) the class of all
n-vertex trees with maximum degree at most \Delta. In this work, we show that
every n-vertex graph satisfying certain natural expansion properties is
T(n,\Delta)-universal or, in other words, contains every spanning tree of
maximum degree at most \Delta. Our methods also apply to the case when \Delta
is some function of n. The result has a few very interesting implications. Most
importantly, we obtain that the random graph G(n,p) is asymptotically almost
surely (a.a.s.) universal for the class of all bounded degree spanning (i.e.,
n-vertex) trees provided that p \geq c n^{-1/3} \log^2n where c > 0 is a
constant. Moreover, a corresponding result holds for the random regular graph
of degree pn. In fact, we show that if \Delta satisfies \log n \leq \Delta \leq
n^{1/3}, then the random graph G(n,p) with p \geq c \Delta n^{-1/3} \log n and
the random r-regular n-vertex graph with r \geq c\Delta n^{2/3} \log n are
a.a.s. T(n,\Delta)-universal. Another interesting consequence is the existence
of locally sparse n-vertex T(n,\Delta)-universal graphs. For constant \Delta,
we show that one can (randomly) construct n-vertex T(n,\Delta)-universal graphs
with clique number at most five. Finally, we show robustness of random graphs
with respect to being universal for T(n,\Delta) in the context of the
Maker-Breaker tree-universality game.Comment: 25 page
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
Hamilton Cycles in Random Subgraphs of Pseudo-Random Graphs
Given an r-regular graph G on n vertices with a Hamilton cycle, order its edges randomly and insert them one by one according to the chosen order, starting from the empty graph. We prove that if the eigenvalue of the adjacency matrix of G with the second largest absolute value satises = o(r 5=2 =(n 3=2 (log n) 3=2 )), then for almost all orderings of the edges of G at the very moment when all degrees of the obtained random subgraph H of G become at least two, H has a Hamilton cycle. As a consequence we derive the value of the threshold for the appearance of a Hamilton cycle in a random subgraph of a pseudo-random graph G, satisfying the above stated condition. Key-words Pseudo-random Graphs, Hamilton Cycles, Random Graphs. 1 Introduction Pseudo-random graphs (sometimes also called quasi-random graphs) can be informally dened as graphs whose edge distribution resembles closely that of truly random graphs on the same number of vertices and with the same edge ..