Louis Esperet

Acyclic edge-coloring using entropy compression

An edge-coloring of a graph G is acyclic if it is a proper edge-coloring of G and every cycle contains at least three colors. We prove that every graph with maximum degree Delta has an acyclic edge-coloring with at most 4 Delta - 4 colors, improving the previous bound of 9.62 (Delta - 1). Our bound results from the analysis of a very simple randomised procedure using the so-called entropy compression method. We show that the expected running time of the procedure is O(mn Delta^2 log Delta), where n and m are the number of vertices and edges of G. Such a randomised procedure running in expected polynomial time was only known to exist in the case where at least 16 Delta colors were available. Our aim here is to make a pedagogic tutorial on how to use these ideas to analyse a broad range of graph coloring problems. As an application, also show that every graph with maximum degree Delta has a star coloring with 2 sqrt(2) Delta^{3/2} + Delta colors.

Exponentially many perfect matchings in cubic graphs

We show that every cubic bridgeless graph G has at least 2|V(G)|/3656 perfect matchings. This confirms an old conjecture of Lovasz and Plummer.

Fire containment in planar graphs

In a graph G, a fire starts at some vertex. At every time step, firefighters can protect up to k vertices, and then the fire spreads to all unprotected neighbors. The k-surviving rate of G is the expectation of the proportion of vertices that can be saved from the fire, if the starting vertex of the fire is chosen uniformly at random. For a given class of graphs , we are interested in the minimum value k such that for some constant and all , (i.e., such that linearly many vertices are expected to be saved in every graph from ). In this note, we prove that for planar graphs this minimum value is at most 4, and that it is precisely 2 for triangle-free planar graphs.

On induced-universal graphs for the class of bounded-degree graphs

For a family F of graphs, a graph U is said to be F-induced-universal if every graph of F is an induced subgraph of U. We give a construction for an induced-universal graph for the family of graphs on n vertices with degree at most k. For k even, our induced-universal graph has O(n^k^/^2) vertices and for k odd it has O(n^@?^k^/^2^@?^-^1^/^klog^2^+^2^/^kn) vertices. This construction improves a result of Butler by a multiplicative constant factor for the even case and by almost a multiplicative n^1^/^k factor for the odd case. We also construct induced-universal graphs for the class of oriented graphs with bounded incoming and outgoing degree, slightly improving another result of Butler.

Coloring planar graphs with three colors and no large monochromatic components

We prove the existence of a function

ISLANDS IN GRAPHS ON SURFACES

f :\mathbb{N} \to \mathbb{N}

A Complexity Dichotomy for the Coloring of Sparse Graphs

such that the vertices of every planar graph with maximum degree

Acyclic improper colourings of graphs with bounded maximum degree

\Delta

On Cubic Bridgeless Graphs Whose Edge-Set Cannot be Covered by Four Perfect Matchings

can be 3-colored in such a way that each monochromatic component has at most

An improved linear bound on the number of perfect matchings in cubic graphs

f(\Delta)