International audienceLet $f(k)$ be the smallest integer such that every $f(k)$-chromatic digraph contains every oriented tree of order $k$. Burr proved $f(k)\leq (k-1)^2$ in general, and he conjectured $f(k)=2k-2$. Burr also proved that every $(8k-7)$-chromatic digraph contains every antidirected tree. We improve both of Burr's bounds. We show that $f(k)\leq k^2/2-k/2+1$ and that every antidirected tree of order $k$ is contained in every $(5k-9)$-chromatic digraph. We make a conjecture that explains why antidirected trees are easier to handle. It states that if $|E(D)| > (k-2) |V(D)|$, then the digraph $D$ contains every antidirected tree of order $k$. This is a common strengthening of both Burr's conjecture for antidirected trees and the celebrated Erd\H{o}s-Sós Conjecture. The analogue of our conjecture for general trees is false, no matter what function $f(k)$ is used in place of $k-2$. We prove our conjecture for antidirected trees of diameter 3 and present some other evidence for it. Along the way, we show that every acyclic $k$-chromatic digraph contains every oriented tree of order $k$ and suggest a number of approaches for making further progress on Burr's conjecture

International audienceAn orientation of a graph G is proper if two adjacent vertices have different in-degrees. The proper-orientation number − → χ (G) of a graph G is the minimum maximum in-degree of a proper orientation of G. In [1], the authors ask whether the proper orientation number of a planar graph is bounded. We prove that every cactus admits a proper orientation with maximum in-degree at most 7. We also prove that the bound 7 is tight by showing a cactus having no proper orientation with maximum in-degree less than 7. We also prove that any planar claw-free graph has a proper orientation with maximum in-degree at most 6 and that this bound can also be attained

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The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-colouring, that is a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic, cartesian and direct products of two graphs in terms of the Grundy numbers of these graphs. Regarding the lexicographic product, we show that Γ(G) × Γ(H) ≤ Γ(G[H]) ≤ 2 Γ(G)−1 (Γ(H) − 1) + Γ(G) − 1. In addition, we show that if G is a tree or Γ(G) = ∆(G) + 1, then Γ(G[H]) = Γ(G) × Γ(H). We then deduce that for every fixed c ≤ 1, given a graph G, it is CoNP-Complete to decide if Γ(G) ≤ c × χ(G) and it is CoNP-Complete to decide if Γ(G) ≤ c × ω(G). Regarding the cartesian product, we show that there is no upper bound of Γ(G H) as a function of Γ(G) and Γ(H). Nevertheless, we prove that for any fixed graph G, there is a function h G such that, for any graph H, Γ(G H) ≤ h G (Γ(H)). Regarding the direct product, we show that Γ(G × H) ≥ Γ(G) + Γ(H) − 2 and construct for any k some graph G k such that Γ(G k) = 2k + 1 and Γ(G k × K 2) = 3k + 1.

An orientation of a graph G is proper if any two adjacent vertices have different indegrees. The proper orientation number − → χ (G) of a graph G is the minimum of the maximum indegree, taken over all proper orientations of G. In this paper, we show that a connected bipartite graph may be properly oriented even if we are only allowed to control the orientation of a specific set of edges, namely, the edges of a spanning tree and all the edges incident to one of its leaves. As a consequence of this result, we prove that 3-connected planar bipartite graphs have proper orientation number at most 6. Additionally, we give a short proof that − → χ (G) ≤ 4, when G is a tree and this proof leads to a polynomial-time algorithm to proper orient trees within this bound.

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a b s t r a c tA b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbour in all other color classes. The b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph H of G. A graph is minimally b-imperfect if it is not b-perfect and every proper induced subgraph is b-perfect. We give a list F of minimally b-imperfect graphs, conjecture that a graph is b-perfect if and only if it does not contain a graph from this list as an induced subgraph, and prove this conjecture for diamond-free graphs, and graphs with chromatic number at most three.

We study a weighted improper coloring problem motivated by a frequency allocation problem. It consists of associating to each vertex a set of p(v) (weight) distinct colors (frequencies), such that the set of vertices having a given color induces a graph of degree at most k (the case k = 0 corresponds to proper coloring). The objective is to minimize the number of colors. We propose approximation algorithms to compute such a coloring for general graphs. We apply these to obtain good approximation ratio for grid and hexagonal graphs. Furthermore we give exact results for the 2-dimensional grid and the triangular lattice when the weights are all the same.