Hervé Hocquard

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

On strong edge-colouring of subcubic graphs

operatorname {mad}(G)

Adjacent vertex-distinguishing edge coloring of graphs

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

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

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

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

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

Strong edge-colouring and induced matchings

A strong edge-colouring of a graph G is a proper edge-colouring such that every path of length 3 uses three different colours. In this paper we improve some previous results on the strong edge-colouring of subcubic graphs by showing that every subcubic graph with maximum average degree strictly less than 73 (resp. 52, 83, 207) can be strongly edge-coloured with six (resp. seven, eight, nine) colours. These upper bounds are optimal except the one of 83. Also, we prove that every subcubic planar graph without 4-cycles and 5-cycles can be strongly edge-coloured with nine colours.

Strong edge-colouring of sparse planar 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].

Graphs with maximum degree 6 are acyclically 11-colorable

Abstract 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 mad ( G ) and Δ ( G ) denote the maximum average degree and the maximum degree of a graph G, respectively. In this note, we prove that every graph G with Δ ( G ) ⩾ 5 and mad ( G ) 13 5 can be avd-colored with Δ ( G ) + 1 colors. This strengthens a result of Wang and Wang [W. Wang and Y. Wang, Adjacent vertex distinguishing edge-colorings of graphs with smaller maximum average degree, J. Comb. Optim., 19:471–485, 2010].

A note on the acyclic 3-choosability of some planar graphs

An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G. A list assignment of G is a function L that assigns to each vertex v@?V(G) a list L(v) of available colors. Let G be a graph and L be a list assignment of G. The graph G is acyclically L-list colorable if there exists an acyclic coloring @f of G such that @f(v)@?L(v) for all v@?V(G). If G is acyclically L-list colorable for any list assignment L with |L(v)|>=k for all v@?V(G), then G is acyclically k-choosable. In this paper, we prove that every planar graph without cycles of lengths 4 to 12 is acyclically 3-choosable.

Strong edge coloring sparse graphs

A strong edge-colouring of a graph G is a proper edge-colouring such that every path of three edges uses three colours. An induced matching of a graph G is a subset I of edges of G such that the graph induced by the endpoints of I is a matching. In this paper, we prove the NP-completeness of strong 4-, 5-, and 6-edge-colouring and maximum induced matching in some subclasses of subcubic triangle-free planar graphs. We also obtain a tight upper bound for the minimum number of colours in a strong edge-colouring of outerplanar graphs as a function of the maximum degree.