Mickaël Montassier

Acyclic 4-Choosability of Planar Graphs Without Cycles of Specific Lengths

A proper vertex coloring of a graph G = (V, E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L-list colorable if for a given list assignment L = {L(v): v ∈ V}, there exists a proper acyclic coloring c of G such that c(v) ∈ L(v) for all v ∈ V. If G is acyclically L-list colorable for any list assignment with |L(v)| ≥ k for all v ∈ V, then G is acyclically k-choosable.

Planar graphs without 5- and 7-cycles and without adjacent triangles are 3-colorable

It is known that planar graphs without cycles of length from 4 to 7 are 3-colorable [O.V. Borodin, A.N. Glebov, A. Raspaud, M.R. Salavatipour, Planar graphs without cycles of length from 4 to 7 are 3-colorable, J. Combin. Theory Ser. B 93 (2005) 303-311]. We improve this result by proving that every planar graph without 5- and 7-cycles and without adjacent triangles is 3-colorable. Also, we give counterexamples to the proof of the same result in [B. Xu, On 3-colorable plane graphs without 5- and 7-cycles, J. Combin. Theory Ser. B 96 (2006) 958-963].

Adjacent vertex-distinguishing edge coloring of graphs with maximum degree Δ

An adjacent vertex-distinguishing edge coloring, or avd-coloring, of a graph G is a proper edge coloring of G such that no pair of adjacent vertices meets the same set of colors. Let

Generalized power domination of graphs

\operatorname {mad}(G)

Acyclic 4-choosability of Planar Graphs with Girth at Least 5

and Δ(G) denote the maximum average degree and the maximum degree of a graph G, respectively. In this paper, we prove that every graph G with Δ(G)≥5 and

Generalized Power Domination in Regular Graphs

\operatorname{mad}(G) < 3-\frac {2}{\Delta}

On strong edge-colouring of subcubic graphs

can be avd-colored with Δ(G)+1 colors. This completes a result of Wang and Wang (J. Comb. Optim. 19:471–485, 2010).

Bordeaux 3-color conjecture and 3-choosability

In this paper, we introduce the concept of k-power domination which is a common generalization of domination and power domination. We extend several known results for power domination to k-power domination. Concerning the complexity of the k-power domination problem, we first show that deciding whether a graph admits a k-power dominating set of size at most t is NP-complete for chordal graphs and for bipartite graphs. We then give a linear algorithm for the problem on trees. Finally, we propose sharp upper bounds for the power domination number of connected graphs and of connected claw-free (k+2)-regular graphs.

Adjacent vertex-distinguishing edge coloring of graphs with maximum degree at least five

A proper vertex coloring of a graph G = (V,E) is acyclic if G contains no bicolored cycle. A graph G is L-list colorable, for a given list assignment L = L(v): v ∈ V, if there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V . If G is L-list colorable for every list assignment with |L(v)| ≥ κ for all v ∈ V, then G is called κ-choosable. A graph is said to be acyclically κ-choosable if these L-list colorings can be chosen to be acyclic. In this paper, we prove that if G is planar with girth g ≥ 5, then G is acyclically 4-choosable. This improves the result of Borodin, Kostochka and Woodall [[BKW99]] concerning the acyclic chromatic number of planar graphs with girth at least 5.

Every planar graph without cycles of lengths 4 to 12 is acyclically 3-choosable

In this paper, we continue the study of power domination in graphs (see [T. W. Haynes et al., SIAM J. Discrete Math., 15 (2002), pp. 519--529; P. Dorbec et al., SIAM J. Discrete Math., 22 (2008), pp. 554--567; A. Aazami et al., SIAM J. Discrete Math., 23 (2009), pp. 1382--1399]). Power domination in graphs was birthed from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A set of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set following a set of rules (according to Kirschoff laws) for power system monitoring. The minimum cardinality of a power dominating set of a graph is its power domination number. We show that the power domination of a connected cubic graph on