Section: New Results
Participants : Nathann Cohen, Frédéric Havet.
Steinberg-like Theorems for Backbone Colouring
Motivated by some channel assignment problem, we study the following variation of graph colouring problem. A function is a (proper) -colouring of G if , for every edge . The chromatic number is the smallest integer for which there exists a proper -colouring of . Given a graph and a subgraph of , a circular -backbone -colouring of is a -colouring of such that , for each edge . The circular -backbone chromatic number of a graph pair , denoted , is the minimum such that admits a circular -backbone -colouring. In  , we first show that if is a planar graph containing no cycle on 4 or 5 vertices and is a forest, then . Then, we prove that if is a forest whose connected components are paths, then .
Complexity of Greedy Edge-colouring
The Grundy index of a graph is the greatest number of colours that the greedy edge-colouring algorithm can use on . In  , we prove that the problem of determining the Grundy index of a graph 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 or and present a polynomial-time algorithm to determine it exactly.
Proper Orientation Number
An orientation of a graph is a digraph obtained from by replacing each edge by exactly one of the two possible arcs with the same endvertices. For each , the indegree of in , denoted by , is the number of arcs with head in . An orientation of is proper if , for all . The proper orientation number of a graph , denoted by , is the minimum of the maximum indegree over all its proper orientations. It is well-known that , for every graph , where and denotes the chromatic number and the maximum degree of . 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  , 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 and such that .
As an evidence to this, we prove that if is bipartite (i.e. ) then .
However, the proper orientation number has the drawback to be difficult to compute. We prove in  that deciding whether is already an NP-complete problem on graphs with , for every . We also show that it is NP-complete to decide whether , for planar subcubic graphs . Moreover, we prove that it is NP-complete to decide whether , for planar bipartite graphs 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 , the proper orientation number is bounded by a function of , this would imply that finding the proper orientation number of a graph with bounded treewidth is polynomial-time solvable. In  we prove if is a tree (or equivalently a graph with treewidth at most 1). In  , we study the cacti which is a special class of graphs with treewidth at most 2. We prove that 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 , deciding whether a graph contains a subdivision of as a subgraph can be solved in polynomial time.
We consider the directed analogue -subdivision problem, which is an analogue for directed graphs (i.e. digraphs). Given a directed graph , does it contain a subdivision of a prescribed digraph ? In  , we give a number of examples of polynomial instances, several NP-completeness proofs as well as a number of conjectures and open problems. In  , we give further support to several open conjectures and speculations about algorithmic complexity of finding -subdivisions. In particular, up to 5 exceptions, we completely classify for which 4-vertex digraphs , the -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.