Po-Shen Loh

Maximizing the number of q-colorings

Let PG(q) denote the number of proper q-colorings of a graph G. This function, called the chromatic polynomial of G, was introduced by Birkhoff in 1912, who sought to attack the famous four-color problem by minimizing PG(4) over all planar graphs G. Since then, motivated by a variety of applications, much research was done on minimizing or maximizing PG(q )o ver various families of graphs. In this paper, we study an old problem of Linial and Wilf, to find the graphs with n vertices and m edges which maximize the number of q-colorings. We provide the first approach that enables one to solve this problem for many nontrivial ranges of parameters. Using our machinery, we show that for each q 4 and sufficiently large m<κ qn 2 ,w hereκq ≈ 1/(q log q), the extremal graphs are complete bipartite graphs minus the edges of a star, plus isolated vertices. Moreover, for q = 3, we establish the structure of optimal graphs for all large m n 2 /4, confirming (in a stronger form) a conjecture of Lazebnik from 1989.

Self-Similarity of Graphs

An old problem raised independently by Jacobson and Schonheim seeks to determine the maximum

Anarchy Is Free in Network Creation

s

Judicious partitions of directed graphs

for which every graph with

Maximizing the Number of Independent Sets of a Fixed Size

m

The critical window for the classical Ramsey-Turán problem

edges contains a pair of edge-disjoint isomorphic subgraphs with

Cops and robbers on planar directed graphs

s

Thresholds for Extreme Orientability

edges. In this paper we determine this maximum up to a constant factor. We show that every

Constrained ramsey numbers

m

On the strong chromatic number of random graphs

-edge graph contains a pair of edge-disjoint isomorphic subgraphs with at least