Daniel M. Martin

Intersecting longest paths

Abstract In 1966, Gallai asked whether every connected graph has a vertex that is common to all longest paths. The answer to this question is negative. We prove that the answer is positive for outerplanar graphs and 2-trees. Another related question was raised by Zamfirescu in the 1980s: Do any three longest paths in a connected graph have a vertex in common? The answer to this question is unknown. We prove that for connected graphs in which all nontrivial blocks are Hamiltonian the answer is affirmative. Finally, we state a conjecture and explain how it relates to the three longest paths question.

A Deterministic Algorithm for the Frieze-Kannan Regularity Lemma

The Frieze-Kannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to efficiently construct a partition satisfying the conditions of the lemma. R. Williams recently asked if one can construct a partition satisfying the conditions of the Frieze-Kannan regularity lemma in deterministic subcubic time. We resolve this problem by designing an

An Optimal Algorithm for Finding Frieze-Kannan Regular Partitions

\tilde O(n^{\omega})

Intersection of Longest Paths in a Graph

time algorithm for constructing such a partition, where

On a conjecture of Thomassen concerning subgraphs of large girth

\omega < 2.376

A deterministic algorithm for the Frieze-Kannan regularity lemma

is the exponent of fast matrix multiplication. The algorithm relies on a spectral characterization of vertex partitions satisfying the properties of the Frieze-Kannan regularity lemma.

On Bichromatic Triangle Game

In this paper we prove that two local conditions involving the degrees and co-degrees in a graph can be used to determine whether a given vertex partition is Frieze{Kannan-regular. With a more rened version of these two local conditions we provide a deterministic algorithm that obtains a Frieze{Kanan-regular partition of any graph G in time O(jV (G)j 2 ).

On Distance Between Graphs

Abstract In 1966, Gallai asked whether every connected graph has a vertex that is common to all its longest paths. The answer to this question is negative. We prove that the answer is positive for outerplanar graphs. Another related question was raised in 1995 at the British Combinatorial Conference: Do any three longest paths in a connected graph have a vertex in common? We prove that, in a connected graph in which all non-trivial blocks are Hamiltonian, any three of its longest paths have a common vertex. Both of these results strengthen a recent result by Axenovich.

Bichromatic Triangle Games

In 1983 C. Thomassen conjectured that for every k, g∈ℕ there exists d such that any graph with average degree at least d contains a subgraph with average degree at least k and girth at least g. Kuhn and Osthus [2004] proved the case g = 6. We give another proof for the case g = 6 which is based on a result of Furedi [1983] about hypergraphs. We also show that the analogous conjecture for directed graphs is true.

Rainbow paths

The Frieze-Kannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to efficiently construct a partition satisfying the conditions of the lemma. Williams [24] recently asked if one can construct a partition satisfying the conditions of the Frieze-Kannan regularity lemma in deterministic sub-cubic time. We resolve this problem by designing an O(nω) time algorithm for constructing such a partition, where ω < 2.376 is the exponent of fast matrix multiplication. The algorithm relies on a spectral characterization of vertex partitions satisfying the properties of the Frieze-Kannan regularity lemma.