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 $f:V\left(G\right)\to \{1,\cdots ,k\}$ is a (proper) $k$-colouring of G if $\left|f\right(u\left)-f\right(v\left)\right|\ge 1$, for every edge $uv\in E\left(G\right)$. The chromatic number $\chi \left(G\right)$ 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 $q\le \left|c\right(u\left)-c\right(v\left)\right|\le k-q$, for each edge $uv\in E\left(H\right)$. The circular $q$-backbone chromatic number of a graph pair $(G,H)$, denoted $CB{C}_q(G,H)$, 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 $H\subseteq G$ is a forest, then $CB{C}_2(G,H)\le 7$. Then, we prove that if $H\subseteq G$ is a forest whose connected components are paths, then $CB{C}_2(G,H)\le 6$.

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 $\Delta \left(G\right)$ or $\Delta \left(G\right)+1$ 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 $v\in V\left(G\right)$, the indegree of $v$ in $D$, denoted by $d_D^{-}\left(v\right)$, is the number of arcs with head $v$ in $D$. An orientation $D$ of $G$ is proper if $d_D^{-}\left(u\right)\ne d_D^{-}\left(v\right)$, for all $uv\in E\left(G\right)$. The proper orientation number of a graph $G$, denoted by $\overrightarrow{\chi }\left(G\right)$, is the minimum of the maximum indegree over all its proper orientations. It is well-known that $\chi \left(G\right)\le \overrightarrow{\chi }\left(G\right)+1\le \Delta \left(G\right)+1$, for every graph $G$, where $\chi \left(G\right)$ and $\Delta \left(G\right)$ 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 $\overrightarrow{\chi }\left(G\right)\le ϵ·\chi \left(G\right)+(1-ϵ)\Delta \left(G\right)$.

As an evidence to this, we prove that if $G$ is bipartite (i.e. $\chi \left(G\right)\le 2$) then $\overrightarrow{\chi }\left(G\right)\le \left(\Delta \left(G\right)+\sqrt{\Delta \left(G\right)}\right)/2+1$.

However, the proper orientation number has the drawback to be difficult to compute. We prove in [17] that deciding whether $\overrightarrow{\chi }\left(G\right)\le \Delta \left(G\right)-1$ is already an NP-complete problem on graphs with $\Delta \left(G\right)=k$, for every $k\ge 3$. We also show that it is NP-complete to decide whether $\overrightarrow{\chi }\left(G\right)\le 2$, for planar subcubic graphs $G$. Moreover, we prove that it is NP-complete to decide whether $\overrightarrow{\chi }\left(G\right)\le 3$, 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 $\overrightarrow{\chi }\left(G\right)\le 4$ 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 $\overrightarrow{\chi }\left(G\right)\le 7$ 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.