Monochromatic cycle covers in random graphs

A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every coloring of the edges of $K_n$ with $r$ colors, there is a cover of its vertex set by at most $f(r) = O(r^2 \log r)$ vertex-disjoint monochromatic cycles. In particular, the minimum number of such covering cycles does not depend on the size of $K_n$ but only on the number of colors. We initiate the study of this phenomena in the case where $K_n$ is replaced by the random graph $\mathcal G(n,p)$. Given a fixed integer $r$ and $p =p(n) \ge n^{-1/r + \varepsilon}$, we show that with high probability the random graph $G \sim \mathcal G(n,p)$ has the property that for every $r$-coloring of the edges of $G$, there is a collection of $f'(r) = O(r^8 \log r)$ monochromatic cycles covering all the vertices of $G$. Our bound on $p$ is close to optimal in the following sense: if $p\ll (\log n/n)^{1/r}$, then with high probability there are colorings of $G\sim\mathcal G(n,p)$ such that the number of monochromatic cycles needed to cover all vertices of $G$ grows with $n$.Comment: 24 pages, 1 figure (minor changes, added figure

Degree-doubling graph families

Let G be a family of n-vertex graphs of uniform degree 2 with the property that the union of any two member graphs has degree four. We determine the leading term in the asymptotics of the largest cardinality of such a family. Several analogous problems are discussed.Comment: 9 page

Finite type coarse expanding conformal dynamics

We continue the study of non-invertible topological dynamical systems with expanding behavior. We introduce the class of {\em finite type} systems which are characterized by the condition that, up to rescaling and uniformly bounded distortion, there are only finitely many iterates. We show that subhyperbolic rational maps and finite subdivision rules (in the sense of Cannon, Floyd, Kenyon, and Parry) with bounded valence and mesh going to zero are of finite type. In addition, we show that the limit dynamical system associated to a selfsimilar, contracting, recurrent, level-transitive group action (in the sense of V. Nekrashevych) is of finite type. The proof makes essential use of an analog of the finiteness of cone types property enjoyed by hyperbolic groups.Comment: Updated versio

The Lifting Properties of A-Homotopy Theory

In classical homotopy theory, two spaces are homotopy equivalent if one space can be continuously deformed into the other. This theory, however, does not respect the discrete nature of graphs. For this reason, a discrete homotopy theory that recognizes the difference between the vertices and edges of a graph was invented, called A-homotopy theory [1-5]. In classical homotopy theory, covering spaces and lifting properties are often used to compute the fundamental group of the circle. In this paper, we develop the lifting properties for A-homotopy theory. Using a covering graph and these lifting properties, we compute the fundamental group of the 5-cycle , giving an alternate approach to [4].Comment: 27 pages, 3 figures, updated version. Minor changes to the introduction and clarification that the computation of the fundamental group of the 5-cycle originally appeared in [4]. Title changed from "Computing A-Homotopy Groups Using Coverings and Lifting Properties" to "The Lifting Properties of A-Homotopy Theory