Carlos Hoppen

Local Algorithms, Regular Graphs of Large Girth, and Random Regular Graphs

We introduce a general class of algorithms and analyse their application to regular graphs of large girth. In particular, we can transfer several results proved for random regular graphs into (deterministic) results about all regular graphs with sufficiently large girth. This reverses the usual direction, which is from the deterministic setting to the random one. In particular, this approach enables, for the first time, the achievement of results equivalent to those obtained on random regular graphs by a powerful class of algorithms which contain prioritised actions. As a result, we obtain new upper or lower bounds on the size of maximum independent sets, minimum dominating sets, maximum k-independent sets, minimum k-dominating sets and maximum k-separated matchings in r-regular graphs with large girth.

CHARACTERIZING TREES WITH LARGE LAPLACIAN ENERGY

We investigate the problem of ordering trees according to their Laplacian energy. More precisely, given a positive integer n, we find a class of cardinality approximately p n whose elements are the n-vertex trees with largest Laplacian energy. The main tool for establishing this result is a new upper bound on the sum Sk(T) of the k largest Laplacian eigenvalues of an n-vertex tree T with diameter at least four, where k 2 f1;:::;ng.

Edge-colorings avoiding a fixed matching with a prescribed color pattern

We consider an extremal problem motivated by a question of Erd?s and Rothschild (Erd?s, 1974) regarding edge-colorings of graphs avoiding a given monochromatic subgraph. An extension of this problem to edge-colorings avoiding fixed subgraphs with a prescribed coloring has been studied by Balogh (Balogh, 2006). In this work, we consider the following natural generalization of the original Erd?s-Rothschild question: given a natural number r and a graph F , an r -pattern P of F is a partition of the edge set of F into r (possibly empty) classes, and an r -coloring of the edge set of a graph G is said to be ( F , P ) -free if it does not contain a copy of F in which the partition of the edge set induced by the coloring has a copy of P . Let c r , ( F , P ) ( G ) be the number of ( F , P ) -free r -colorings of a graph G . For large n , we maximize this number over all n -vertex graphs for a large class of patterns in matchings and we describe the graphs that achieve this maximum.

Edge-colorings of graphs avoiding complete graphs with a prescribed coloring

Given a graph

A note on permutation regularity

F

Edge-colorings avoiding fixed rainbow stars☆

and an integer

A rainbow Erdős-Rothschild problem

r \ge 2

A Rainbow Erdös--Rothschild Problem

, a partition

A coloring problem for intersecting vector spaces

\widehat{F}

Unicyclic graphs with equal Laplacian energy

of the edge set of