Alexandre Pinlou

The Domination Number of Grids

In this paper, we conclude the calculation of the domination number of all (n,m) grid graphs. Indeed, we prove Chang’s conjecture saying that for every 16≤n≤m, γ(Gn,m)=⌊(n+2)(m+2)5⌋-4.

Triangle Contact Representations and Duality

Axa0contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. De Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles. We strengthen this in terms of a simultaneous contact representation by triangles of a planar map and of its dual.Axa0primal–dual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv, bordering faces f and g, the intersection between the triangles corresponding to u and v is the same point as the intersection between the triangles corresponding to f and g. We prove that every 3-connected planar map admits a primal–dual contact representation by triangles. Moreover, the interiors of the triangles form a tiling of the triangle corresponding to the outer face and each contact point is a corner of exactly three triangles. Then we show that these representations are in one-to-one correspondence with generalized Schnyder woods defined by Felsner for 3-connected planar maps.

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.

Oriented colorings of partial 2-trees

A homomorphism from an oriented graph G to an oriented graph H is an arc-preserving mapping f from V(G) to V(H), that is f(x)f(y) is an arc in H whenever xy is an arc in G. The oriented chromatic number of G is the minimum order of an oriented graph H such that G has a homomorphism to H. In this paper, we determine the oriented chromatic number of the class of partial 2-trees for every girth g>=3. We also give an upper bound for the oriented chromatic number of planar graphs with girth at least 11.

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.

An oriented coloring of planar graphs with girth at least five

An oriented k-coloring of an oriented graph G is a homomorphism from G to an oriented graph H of order k. We prove that every oriented graph with a maximum average degree less than 103 and girth at least 5 has an oriented chromatic number at most 16. This implies that every oriented planar graph with girth at least 5 has an oriented chromatic number at most 16, that improves the previous known bound of 19 due to Borodin et al. [O.V. Borodin, A.V. Kostochka, J. Nesetril, A. Raspaud, E. Sopena, On the maximum average degree and the oriented chromatic number of a graph, Discrete Math. 206 (1999) 77-89].

Triangle contact representations and duality

A contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. de Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles. We strengthen this in terms of a simultaneous contact representation by triangles of a planar map and of its dual. n nA primal-dual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv, bordering faces f and g, the intersection between the triangles corresponding to u and v is the same point as the intersection between the triangles corresponding to f and g. We prove that every 3-connected planar map admits a primal-dual contact representation by triangles. Moreover, the interiors of the triangles form a tiling of the triangle corresponding to the outer face and each contact point is a node of exactly three triangles. Then we show that these representations are in one-to-one correspondence with generalized Schnyder woods defined by Felsner for 3-connected planar maps.

A Complexity Dichotomy for the Coloring of Sparse Graphs

Galluccio, Goddyn, and Hell proved in 2001 that in any minor-closed class of graphs, graphs with large enough girth have a homomorphism to any given odd cycle. In this paper, we study the computational aspects of this problem. Let be a monotone class of graphs containing all planar graphs, and closed under clique-sum of order at most two. Examples of such class include minor-closed classes containing all planar graphs, and such that all minimal obstructions are 3-connected. We prove that for any k and g, either every graph of girth at least g in has a homomorphism to , or deciding whether a graph of girth g in has a homomorphism to is NP-complete. We also show that the same dichotomy occurs when considering 3-Colorability or acyclic 3-Colorability of graphs under various notions of density that are related to a question of Havel (On a conjecture of Grunbaum, J Combin Theory Ser B 7 (1969), 184–186) and a conjecture of Steinberg (The state of the three color problem, Quo Vadis, Graph theory?, Ann Discrete Math 55 (1993), 211–248) about the 3-Colorability of sparse planar graphs.

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.

Homomorphisms of 2-edge-colored graphs

In this paper, we study homomorphisms of 2-edge-colored graphs, that is graphs with edges colored with two colors. We consider various graph classes (outerplanar graphs, partial 2-trees, partial 3-trees, planar graphs) and the problem is to find, for each class, the smallest number of vertices of a 2-edge-colored graph H such that each graph of the considered class admits a homomorphism to H.