Jonathan A. Noel

A Proof of a Conjecture of Ohba

We prove a conjecture of Ohba that says that every graph G on at most 2i¾?G+1 vertices satisfies i¾?i¾?G=i¾?G.

Mixing Homomorphisms, Recolorings, and Extending Circular Precolorings

This work brings together ideas of mixing graph colorings, discrete homotopy, and precoloring extension. A particular focus is circular colorings. We prove that all the k,q-colorings of a graph G can be obtained by successively recoloring a single vertex provided k/qi¾?2colG along the lines of Cereceda, van den Heuvel, and Johnsons result for k-colorings. We give various bounds for such mixing results and discuss their sharpness, including cases where the bounds for circular and classical colorings coincide. As a corollary, we obtain an Albertson-type extension theorem for k,q-precolorings of circular cliques. Such a result was first conjectured by Albertson and West. General results on homomorphism mixing are presented, including a characterization of graphs G for which the endomorphism monoid can be generated through the mixing process. As in similar work of Brightwell and Winkler, the concept of dismantlability plays a key role.

Saturation in the Hypercube and Bootstrap Percolation

Let Qd denote the hypercube of dimension d. Given d ≥ m, a spanning subgraph G of Qd is said to be (Qd,Qm)-saturated if it does not contain Qm as a subgraph but adding any edge of E(Qd) \ E(G) creates a copy of Qm in G. Answering a question of Johnson and Pinto [27], we show that for every fixed m ≥ 2 the minimum number of edges in a (Qd,Qm)-saturated graph is �(2 d ). We also study weak saturation, which is a form of bootstrap percolation. Given graphs F and H, a spanning subgraph G of F is said to be weakly (F,H)-saturated if the edges of E(F)\E(G) can be added to G one at a time so that each additional edge creates a new copy of H. Answering another question of Johnson and Pinto [27], we determine the minimum number of edges in a weakly (Qd,Qm)-saturated graph for all d ≥ m ≥ 1. More generally, we determine the minimum number of edges in a subgraph of the d-dimensional grid P d k which is weakly saturated with respect to ‘axis aligned’ copies of a smaller grid P m r . We also study weak saturation of cycles in the grid.

Choosability of Graphs with Bounded Order: Ohba's Conjecture and Beyond

Abstract We discuss some recent results and conjectures on bounding the choice number of a graph G under the condition that | V ( G ) | is bounded above by a fixed function of χ ( G ) .