Marthe Bonamy

Recoloring bounded treewidth graphs

Abstract Let k be an integer. Two vertex k-colorings of a graph are adjacent if they differ on exactly one vertex. A graph is k-mixing if any proper k-coloring can be transformed into any other through a sequence of adjacent proper k-colorings. Any graph is ( t w + 2 ) -mixing, where tw is the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two ( t w + 2 ) -colorings is at most quadratic, a problem left open in Bonamy et al. (2012). Jerrum proved that any graph is k-mixing if k is at least the maximum degree plus two. We improve Jerrumʼs bound using the grundy number, which is the worst number of colors in a greedy coloring.

Graphs with maximum degree Δ≥17 and maximum average degree less than 3 are list 2-distance (Δ+2)-colorable

For graphs of bounded maximum average degree, we consider the problem of 2-distance coloring. This is the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. It is already known that planar graphs of girth at least 6 and of maximum degree @D are list 2-distance (@D+2)-colorable when @D>=24 (Borodin and Ivanova (2009)) and 2-distance (@D+2)-colorable when @D>=18 (Borodin and Ivanova (2009)). We prove here that @D>=17 suffices in both cases. More generally, we show that graphs with maximum average degree less than 3 and @D>=17 are list 2-distance (@D+2)-colorable. The proof can be transposed to list injective (@D+1)-coloring.

On the diameter of reconfiguration graphs for vertex colourings.

The reconfiguration graph of the k-colourings of a graph G contains as its vertex set the proper vertex k-colourings of G, and two colourings are joined by an edge in the reconfiguration graph if they differ in colour on just one vertex of G. We prove that for a graph G on n vertices that is chordal or chordal bipartite, if G is k-colourable, then the reconfiguration graph of its l-colourings, for l⩾k+1, is connected and has diameter O(n2). We show that this bound is asymptotically tight up to a constant factor.

2‐Distance Coloring of Sparse Graphs

A 2-distance coloring of a graph is a coloring of the vertices such that two vertices at distance at most 2 receive distinct colors. We prove that every graph with maximum degree Δ at least 4 and maximum average degree less that 7 admits a 2-distance (Δ + 1)-coloring. This result is tight. This improves previous known results of Dolama and Sopena.

Adjacent vertex-distinguishing edge coloring of graphs

An adjacent vertex-distinguishing edge coloring (AVD-coloring) of a graph is a proper edge coloring such that no two neighbors are adjacent to the same set of colors. Zhang et al. [17] conjectured that every connected graph on at least 6 vertices is AVD (Δ + 2)-colorable, where A is the maximum degree. In this paper, we prove that (Δ + 1) colors are enough when A is sufficiently larger than the maximum average degree, denoted mad. We also provide more precise lower bounds for two graph classes: planar graphs, and graphs with mad < 3. In the first case, Δ ≥ 12 suffices, which generalizes the result of Edwards et al. [7] on planar bipartite graphs. No other results are known in the case of planar graphs. In the second case, Δ ≥ 4 is enough, which is optimal and completes the results of Wang and Wang [14] and of Hocquard and Montassier [9].

2-distance coloring of sparse graphs

Abstract A 2-distance coloring of a graph is a coloring of the vertices such that two vertices at distance at most 2 receive distinct colors. We prove that every graph with maximum degree Δ at least 4 and maximum average degree less that 7 3 admits a 2-distance ( Δ + 1 ) -coloring. This result is tight. This improves previous known results of Dolama and Sopena.

Recoloring graphs via tree decompositions

Let

The Erdös--Hajnal Conjecture for Long Holes and Antiholes

k

List coloring the square of sparse graphs with large degree

be an integer. Two vertex

Recognizing graphs close to bipartite graphs.

k