Georgia Institute of Technology College of Sciences

I will find upper and lower bounds for the mixing time as well as the likelihood order after sufficient time of the following involution walk on the symmetric group. Consider 2n cards on a table. Pair up all the cards. Ignore each pairing with probability $p \geq 1/2$. For any pair not being ignored, pick up the two cards and switch their spots. This walk is generated by involutions with binomially distributed two cycles. The upper bound of $\log_{2/(1+p)}(n)$ will result from spectral analysis using a combination of a series of monotonicity relations on the eigenvalues of the walk and the character polynomial for the representations of the symmetric group. A lower bound of $\log_{1/p}$ differs by a constant factor from the upper bound. This walk was introduced to study a conjecture about a random walk on the unitary group from the information theory of black holes.

A graph is a minor of another graph if the former can be obtained from a subgraph of the latter by contracting edges. We prove that for every graph H, if H is not a minor of a graph G, then V(G) can be 3-colored such that the subgraph induced by each color class has no component with size greater than a function of H and the maximum degree of G. This answers a question raised by Esperet and Joret, generalizes their result for 3-coloring V(G) for graphs G embeddable in a fixed surface, and improves a result of Alon, Ding, Oporowski and Vertigan for 4-coloringing V(G) for H-minor free graphs G. As a corollary, we prove that for every positive integer t, if G is a graph with no K_{t+1} minor, then V(G) can be 3t-colored such that the subgraph induced by each color class has no component with size larger than a function of t. This improves a result of Wood for coloring V(G) by 3.5t+2 colors. This work is joint with Sang-il Oum.

A set F of graphs has the Erdos-Posa property if there exists a function f such that every graph either contains k disjoint subgraphs each isomorphic to a member in F or contains at most f(k) vertices intersecting all such subgraphs. In this talk I will address the Erdos-Posa property with respect to three closely related graph containment relations: minor, topological minor, and immersion. We denote the set of graphs containing H as a minor, topological minor and immersion by M(H),T(H) and I(H), respectively. Robertson and Seymour in 1980's proved that M(H) has the Erdos-Posa property if and only if H is planar. And they left the question for characterizing H in which T(H) has the Erdos-Posa property in the same paper. This characterization is expected to be complicated as T(H) has no Erdos-Posa property even for some tree H. In this talk, I will present joint work with Postle and Wollan for providing such a characterization. For immersions, it is more reasonable to consider an edge-variant of the Erdos-Posa property: packing edge-disjoint subgraphs and covering them by edges. I(H) has no this edge-variant of the Erdos-Posa property even for some tree H. However, I will prove that I(H) has the edge-variant of the Erdos-Posa property for every graph H if the host graphs are restricted to be 4-edge-connected. The 4-edge-connectivity cannot be replaced by the 3-edge-connectivity.