Section: New Results

Graph theory

Participants : Nathann Cohen, Frédéric Havet.

Graph Colouring

Steinberg-like Theorems for Backbone Colouring

Motivated by some channel assignment problem, we study the following variation of graph colouring problem. A function f:V(G){1,,k} is a (proper) k-colouring of G if |f(u)f(v)|1, for every edge uvE(G). The chromatic number χ(G) is the smallest integer k for which there exists a proper k-colouring of G. Given a graph G and a subgraph H of G, a circular q-backbone k-colouring c of (G,H) is a k-colouring of G such that q|c(u)c(v)|kq, for each edge uvE(H). The circular q-backbone chromatic number of a graph pair (G,H), denoted CBCq(G,H), is the minimum k such that (G,H) admits a circular q-backbone k-colouring. In [19] , we first show that if G is a planar graph containing no cycle on 4 or 5 vertices and HG is a forest, then CBC2(G,H)7. Then, we prove that if HG is a forest whose connected components are paths, then CBC2(G,H)6.

Complexity of Greedy Edge-colouring

The Grundy index of a graph G=(V,E) is the greatest number of colours that the greedy edge-colouring algorithm can use on G. In [33] , we prove that the problem of determining the Grundy index of a graph G=(V,E) is NP-hard for general graphs. We also show that this problem is polynomial-time solvable for caterpillars. More specifically, we prove that the Grundy index of a caterpillar is Δ(G) or Δ(G)+1 and present a polynomial-time algorithm to determine it exactly.

Proper Orientation Number

An orientation of a graph G is a digraph D obtained from G by replacing each edge by exactly one of the two possible arcs with the same endvertices. For each vV(G), the indegree of v in D, denoted by dD-(v), is the number of arcs with head v in D. An orientation D of G is proper if dD-(u)dD-(v), for all uvE(G). The proper orientation number of a graph G, denoted by χ(G), is the minimum of the maximum indegree over all its proper orientations. It is well-known that χ(G)χ(G)+1Δ(G)+1, for every graph G, where χ(G) and Δ(G) denotes the chromatic number and the maximum degree of G. In other words, the proper orientation number (plus one) is an upper bound on the chromatic number which is tighter than the maximum degree.

In [17] , we ask whether the proper orientation number is really a more accurate bound than the maximum degree in the following sense : does there exists a positive ϵ and such that χ(G)ϵ·χ(G)+(1-ϵ)Δ(G).

As an evidence to this, we prove that if G is bipartite (i.e. χ(G)2) then χ(G)Δ(G)+Δ(G)/2+1.

However, the proper orientation number has the drawback to be difficult to compute. We prove in [17] that deciding whether χ(G)Δ(G)-1 is already an NP-complete problem on graphs with Δ(G)=k, for every k3. We also show that it is NP-complete to decide whether χ(G)2, for planar subcubic graphs G. Moreover, we prove that it is NP-complete to decide whether χ(G)3, for planar bipartite graphs G with maximum degree 5.

Nevertheless, it might be interesting to bound the proper orientation number on some graph families. In particular, if we prove that for a graph with treewidth at most t, the proper orientation number is bounded by a function of t, this would imply that finding the proper orientation number of a graph with bounded treewidth is polynomial-time solvable. In [17] we prove χ(G)4 if G is a tree (or equivalently a graph with treewidth at most 1). In [53] , we study the cacti which is a special class of graphs with treewidth at most 2. We prove that χ(G)7 for every cactus. We also prove that the bound 7 is tight by showing a cactus having no proper orientation with maximum indegree less than 7. We also prove that any planar claw-free graph has a proper orientation with maximum indegree at most 6 and that this bound can also be attained.

Subdivisions of Digraphs

An important result in the Roberston and Seymour minor theory is the polynomial-time algorithm to solve the so-called Linkage Problem. This implies in particular, that for any fixed graph H, deciding whether a graph G contains a subdivision of H as a subgraph can be solved in polynomial time.

We consider the directed analogue F-subdivision problem, which is an analogue for directed graphs (i.e. digraphs). Given a directed graph D, does it contain a subdivision of a prescribed digraph F? In [20] , we give a number of examples of polynomial instances, several NP-completeness proofs as well as a number of conjectures and open problems. In [62] , we give further support to several open conjectures and speculations about algorithmic complexity of finding F-subdivisions. In particular, up to 5 exceptions, we completely classify for which 4-vertex digraphs F, the F-subdivision problem is polynomial-time solvable and for which it is NP-complete. While all NP-hardness proofs are made by reduction from some version of the 2-linkage problem in digraphs, some of the polynomial-time solvable cases involve relatively complicated algorithms.