Aline Parreau

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.

Extremal graphs for the identifying code problem

An identifying code of a graph G is a dominating set C such that every vertex x of G is distinguished from other vertices by the set of vertices in C that are at distance at most 1 from x. The problem of finding an identifying code of minimum possible size turned out to be a challenging problem. It was proved by N. Bertrand, I. Charon, O. Hudry and A. Lobstein that if a graph on n vertices with at least one edge admits an identifying code, then a minimal identifying code has size at most n-1. They introduced classes of graphs whose smallest identifying code is of size n-1. Few conjectures were formulated to classify the class of all graphs whose minimum identifying code is of size n-1. In this paper, disproving these conjectures, we classify all finite graphs for which all but one of the vertices are needed to form an identifying code. We classify all infinite graphs needing the whole set of vertices in any identifying code. New upper bounds in terms of the number of vertices and the maximum degree of a graph are also provided.

IDENTIFYING CODES IN LINE GRAPHS

An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We study the edge-identifying code problem, i.e. the identifying code problem in line graphs. If

IDENTIFYING CODES IN HEREDITARY CLASSES OF GRAPHS AND VC-DIMENSION ∗

\ID(G)

New results on variants of covering codes in Sierpiński graphs

denotes the size of a minimum identifying code of an identifiable graph

Algorithms and Complexity for Metric Dimension and Location-domination on Interval and Permutation Graphs

G

Locally identifying coloring in bounded expansion classes of graphs

, we show that the usual bound

Identification, location–domination and metric dimension on interval and permutation graphs. I. Bounds

\ID(G)\ge \lceil\log_2(n+1)\rceil

Characterizing Extremal Digraphs for Identifying Codes and Extremal Cases of Bondy's Theorem on Induced Subsets

, where

An improved lower bound for (1,<=2)-identifying codes in the king grid

n