Tien-Nam Le

Edge-Partitioning a Graph into Paths: Beyond the Barát-Thomassen Conjecture

In 2006, Barát and Thomassen conjectured that there is a function f such that, for every fixed tree T with t edges, every f(t)-edge-connected graph with its number of edges divisible by t has a partition of its edges into copies of T. This conjecture was recently verified by the current authors and Merker [1].We here further focus on the path case of the Barát-Thomassen conjecture. Before the aforementioned general proof was announced, several successive steps towards the path case of the conjecture were made, notably by Thomassen [11,12,13], until this particular case was totally solved by Botler, Mota, Oshiro andWakabayashi [2]. Our goal in this paper is to propose an alternative proof of the path case with a weaker hypothesis: Namely, we prove that there is a function f such that every 24-edge-connected graph with minimum degree f(t) has an edge-partition into paths of length t whenever t divides the number of edges. We also show that 24 can be dropped to 4 when the graph is eulerian.

Forcing clique immersions through chromatic number

Abstract Building on recent work of Dvořak and Yepremyan, we show that every simple graph of minimum degree 7 t + 7 contains K t as an immersion and that every graph with chromatic number at least 3.54 t + 4 contains K t as an immersion.

Coloring dense digraphs

The chromatic number of a digraph D is the minimum number of acyclic subgraphs covering the vertex set of D. A tournament H is a hero if every H-free tournament T has chromatic number bounded by a function of H. Inspired by the celebrated Erdős-Hajnal conjecture, Berger et al. fully characterized the class of heroes in 2013. We extend this framework to dense digraphs: A digraph H is a superhero if every H-free digraph D has chromatic number bounded by a function of H and α(D), the independence number of the underlying graph of D. We prove here that a digraph is a superhero if and only if it is a hero, and hence characterize all superheroes. This answers a question of Aboulker, Charbit and Naserasr.

Additive Bases and Flows in Graphs

It was conjectured by Jaeger et al. in 1992 that for any prime number

Additive bases and flows in graphs

p

Every 3-colorable graph has a faithful representation in the odd-distance graph

, there is a constant

Subquadratic kernels for implicit 3-h itting s et and 3-s et p acking problems

c

Coloring tournaments: from local to global

such that for any

A Proof of the Bar\'at-Thomassen Conjecture

n

Domination and Fractional Domination in Digraphs

, the union (with repetition) of the vectors of any family of