Project-Mascotte:Graph Theory

Inria / Raweb 2009
Presentation of the Project Mascotte

Section: New Results

Graph Theory

Participants : Julio Araujo, Jorgen Bang-Jensen, Jean-Claude Bermond, Nathann Cohen, David Coudert, Frédéric Giroire, Frédéric Havet, Nicolas Nisse, Stéphane Pérennes, Bruce Reed, Leonardo Sampaio, Ignasi Sau-Valls.

Mascotte principally investigates applications in telecommunications via Graph Theory (see other objectives). However it also studies a number of theoretical problems of general interest. Our research mainly focused on three important topics: graph colouring, width parameters and random graphs.

Graph colouring is a hot topic in Graph Theory. It is one of the oldest problem in combinatorics (with the 4-colour problem), has a central position in Discrete Mathematics and a huge number of applications. Lots of new results have been obtained the last ten years with the fast development of new techniques (structural and probabilistic). In Mascotte we studied graph colouring problems via these new methods (probabilistic method, discharging method,...).
The theory of width parameters (tree-width, branch-width, ...) is a deep and algorithmically useful structure theory. Therefore it is now widely studied and used. In particular, it is strongly related to graph searching problems (see 6.6 ).
Since the seminal paper of Erdös and Rényi, the theory of random graphs has now grown a very active field with an extensive literature. There are many beautiful results in the theory of random graphs as well as various applications in computer science, biology, ... In Mascotte, we study random graphs for their own sake but as well as tools to solve some graph-theoretic questions which have nothing to do with randomness.

Graph colouring

Colouring and edge-colouring are two central concepts in Graph Theory. There are many important and long standing conjectures in these areas. We are trying to make advances towards such conjectures, in particular Hadwiger's conjecture, the List Colouring Conjecture and the Acyclic Edge-Colouring Conjecture. We also investigated the relation between the chromatic number and the crossing number of a graph.

We are also interested in colouring problems arising from some practical problems: improper colouring, L(p, q) -labelling, directed star arboricity and good edge-labelling. The first two are both motivated by channel assignment and the last two are motivated by problems arising in WDM networks. In [70] , some of these problems are summarized.

We also studied some other variants of colouring like circular colouring, non-repetitive colouring and frugal colouring.

Hadwiger's conjecture: The famous Hadwiger's conjecture asserts that every graph with no K_t -minor is (t-1) -colourable. The case t = 5 is known to be equivalent to the Four Colour Theorem by Wagner, and the case t = 6 is settled by Robertson, Seymour and Thomas. So far the cases t $\ge$ 7 are wide open. In [73] , we prove the following two theorems: There is an O(n²) algorithm to decide whether or not a given graph G satisfies Hadwiger's conjecture for the case t . Every minimal counterexample to Hadwiger's conjecture for the case t has at most f(t) vertices for some explicit bound f(t) .

In [31] , we show an approximate version of Hadwiger's conjecture. A K_t -expansion consists of t vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be two-coloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every t , if a graph contains no odd K_t -expansion then its chromatic number is $Im1 ${O(t\sqrt {logt})}$$ . In doing so, we obtain a characterization of graphs which contain no odd K_t -expansion which is of independent interest. We also prove that given a graph and a subset S of its vertex set, either there are k vertex-disjoint odd paths with endpoints in S , or there is a set X of at most 2k² vertices such that every odd path with both ends in S contains a vertex in X . Finally, we discuss the algorithmic implications of these results.

Edge-colouring: The most celebrated conjecture on edge-colouring is the List Colouring Conjecture asserting that the chromatic index is always equal to the list chromatic index. Together with Vizing's Theorem it implies the following conjecture : For any graph G with maximum degree $\upper_delta$ , the list chromatic index is at most $\upper_delta$ + 1 . In [105] , we give a short proof of a result of Borodin showing that this later conjecture holds for planar graphs of maximum degree at least 9.

We also investigate the algorithmic issue of edge-colouring. For any c>1 , we describe [74] a linear time algorithm for fractionally edge colouring simple graphs with maximum degree at least |V|/c .

A proper edge-colouring with the property that every cycle contains edges of at least three distinct colours is called an acyclic edge-colouring . The acyclic chromatic index of a graph G , denoted $\chi$ _a^'(G) is the minimum k such that G admits an acyclic edge-colouring with k colours. The Acyclic Colouring Conjecture states that $\chi$ _a^'(G) = $\upper_delta$ (G) + 2 for every graph G . In [106] , [56] , we conjecture that if G is planar and $\upper_delta$ (G) is large enough then $\chi$ _a^'(G) = $\upper_delta$ (G) . We settle this conjecture for planar graphs with girth at least 5 and outerplanar graphs. We also show that $\chi$ _a^'(G) $\le$ $\upper_delta$ (G) + 25 for all planar graph G .

Crossing and colouring: The crossing number of a graph G , denoted by cro(G) , is the minimum number of crossings in any drawing of G in the plane.

The Four Colour Theorem states that if a graph has crossing number zero then it is 4-colourable. It is then natural to find upper bounds on the chromatic number in terms of its crossing number. Oporowski and Zhao showed that a graph with crossing number at most 3 is 5-colourable unless it contains a K₆ . They conjecture that this result could be extended to graphs with crossing number at most 5. In [110] , we disprove this conjecture but show that every graph with crossing number at most 4 and containing no K₆ is 5-colourable. We also show some colourability results on graphs that can be made planar by removing few edges. In particular, we show that if there exists three edges whose removal leaves the graph planar then it is 5-colourable.

Improper colouring: A k -improper $\ell$ -colouring is a mapping c from its vertex set into a set of colours such that every vertex has at most k neighbours with the same colour. A result of Lovász states that for any graph G , such a partition exists if $Im2 ${\#8467 \#8805 \mfenced o=⌈ c=⌉ \mfrac {\#916 (G)+1}{k+1}}$$ . When k = 0 , this bound can be reduced by Brooks' Theorem, unless G is complete or an odd cycle. In [28] , we study the following question, which can be seen as a generalisation of the celebrated Brooks' Theorem to improper colouring: does there exist a polynomial-time algorithm that decides whether a graph G of maximum degree $\upper_delta$ has k -improper chromatic number at most $Im3 ${\#8968 \mfrac {\#916 +1}{k+1}\#8969 -1}$$ ? We show that the answer is no, unless P = NP, when $\upper_delta$ = $\ell$ (k + 1) , k $\ge$ 1 and $Im4 ${\#8467 +\sqrt \#8467 \#8804 2k+3}$$ . We also show that, if G is planar, k = 1 or k = 2 , $\upper_delta$ = 2k + 2 , and $\ell$ = 2 , then the answer is still no, unless P = NP. These results answer some questions of Cowen et al. [Journal of Graph Theory 24(3):205-219, 1997].

L(p, q) -labelling: An L(p, q) -labelling of G is an integer assignment f to the vertex set V(G) such that |f(u)-f(v)| $\ge$ p , if u and v are adjacent, and |f(u)-f(v)| $\ge$ q , if u and v have a common neighbour. Such a concept is a modeling of a simple channel assignment, in which the separation between channels depends on the distance. More precisely, it has to be at least p if they are very close and q if they are close (but not very close). The goal is to find an L(p, q) -labelling f of G with minimum span (i.e. max{f(u)-f(v), u, v $\in$ V(G)} ). It is well known that deciding if a graph has an L(p, 1) -labelling with minimum span k is NP-complete. We show that it remains NP-complete when restricted to planar graphs [109] or vertex-edge incidence graphs [36] which form a small class of bipartite graphs. We also give [33] some upper bouns for the span of an L(1, 1) -labelling of a planar graph with large girth.

Directed star arboricity: A star is an arborescence in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. The directed star arboricity of a digraph D , denoted by dst(D) , is the minimum number of galaxies needed to cover A(D) . In [69] , we show that dst(D) $\le$ $\upper_delta$ (D) + 1 and that if D is acyclic then dst(D) $\le$ $\upper_delta$ (D) . These results are proved by considering the existence of spanning galaxies in digraphs. Thus, we study the problem of deciding whether a digraph D has a spanning galaxy or not. We show that it is NP-complete (even when restricted to acyclic digraphs) but that it becomes polynomial-time solvable when restricted to strongly connected digraphs.

Good edge-labelling: Let $Im5 $\#119979 $$ be a family of dipaths of a DAG (Directed Acyclic Graph) G . The load of an arc is the number of dipaths containing this arc. Let $Im6 ${\#960 (G,\#119979 )}$$ be the maximum of the load of all the arcs and let $Im7 ${w(G,\#119979 )}$$ be the minimum number of wavelengths (colours) needed to colour the dipaths of $Im5 $\#119979 $$ in such a way that two dipaths with the same wavelength are arc-disjoint. There exist DAGs such that the ratio between $Im7 ${w(G,\#119979 )}$$ and $Im6 ${\#960 (G,\#119979 )}$$ cannot be bounded. An internal cycle is an oriented cycle such that all the vertices have at least one predecessor and one successor in G (said otherwise every cycle contain neither a source nor a sink of G ). We prove [97] that, for any family of dipaths $Im5 $\#119979 $$ , $Im8 ${w(G,\#119979 )=\#960 (G,\#119979 )}$$ if and only if G has no internal cycle. We also consider a new class of DAGs, which is of interest in itself, those for which there is at most one dipath from a vertex to another. We call these digraphs UPP-DAGs. For these UPP-DAGs we show that the load is equal to the maximum size of a clique of the conflict graph. We prove that the ratio between $Im7 ${w(G,\#119979 )}$$ and $Im6 ${\#960 (G,\#119979 )}$$ cannot be bounded. For that we introduce good edge-labellings of the conflict graph, namely edge-labellings such that for any ordered pair of vertices (x, y) there do not exist two paths from x to y with increasing labels. In [48] , [95] , we aim at characterizing the class of graphs that admit a good edge-labelling. First, we exhibit infinite families of graphs for which no such edge-labelling can be found. We then show that deciding if a graph admits a good edge-labelling is NP-complete. Finally, we give large classes of graphs admitting a good edge-labelling: C₃ -free outerplanar graphs, planar graphs of girth at least 6, subcubic {C₃, K_{2, 3}} -free graphs.

Varied colourings: In [34] , we study circular choosability, a notion recently introduced by Mohar and by Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that $Im9 ${cch{(G)}=O\mfenced o=( c=) ch(G)+ln|V(G)|}$$ for every graph G . We investigate a generalisation of circular choosability, the circular f -choosability, where f is a function of the degrees. We also consider the circular choice number of planar graphs. Mohar asked for the value of $Im10 ${\#964 :=sup{cch(G):G~\mtext is~\mtext planar}}$$ , and we prove that 6 $\le$ $\tau$ $\le$ 8 , thereby providing a negative answer to another question of Mohar. We also study the circular choice number of planar and outerplanar graphs with prescribed girth, and graphs with bounded density.

We also study non-repetitive colouring. A sequence $Im11 ${r_1,r_2,\#8943 ,r_{2n}}$$ such that r_i = r_{n + i} for all 1 $\le$ i $\le$ n , is called a repetition . A sequence S is called non-repetitive if no block (i.e. subsequence of consecutive terms of S ) is a repetition. Let G be a graph whose edges are coloured. A trail is called non-repetitive if the sequence of colours of its edges is non-repetitive. If G is a plane graph, a facial non-repetitive edge-colouring of G is an edge-colouring such that any facial trail (i.e. trail of consecutive edges on the boundary walk of a face) is non-repetitive. We denote $\pi$ _f^'(G) the minimum number of colours of a facial non-repetitive edge-colouring of G . In [112] , we show that $\pi$ _f^'(G) $\le$ 8 for any plane graph G . We also get better upper bounds for $\pi$ _f^'(G) in the cases when G is a tree, a plane triangulation, a simple 3-connected plane graph, a hamiltonian plane graph, an outerplanar graph or a Halin graph. The bound 4 for trees is tight.

We also worked on frugal colouring. In [80] , we prove that every graph with maximum degree $\upper_delta$ can be properly ( $\upper_delta$ + 1) -coloured so that no colour appears more than O(log $\upper_delta$ /loglog $\upper_delta$ ) times in the neighbourhood of any vertex. This is best possible up to the constant factor in the O(–) term. We also provide an efficient algorithm to produce such a colouring.

Finally, in [43] , we consider the class of graphs that contain no odd hole, no antihole, and no ”prism” (a graph consisting of two disjoint triangles with three disjoint paths between them). We give an algorithm that can optimally color the vertices of these graphs in time O(n²m) .

Width parameters

A key notion in the tree-width theory is the duality between the bramble-number of a graph and its tree-width. Adapting the method introduced in Graph Minors x [Robertson and Seymour, Journal of Combinatorial Theory B 52(2): 153-190 (1991)], we propose a new proof of it [23] . Our approach is based on a new definition of submodularity on partition functions which naturally extends the usual one on set functions. The proof does not rely on Menger's theorem, and thus greatly generalises the original one. It thus provides a dual for matroid tree-width. One can also derive all known dual notions of other classical width-parameters from it.

On the algorithmic point of view, lots of polynomial time algorithms based on tree-width use a related duality fact: if a graph has no r×r grid minor then its tree-width is bounded by 2^20r⁵ . This huge upper bound is far from being tight (a polynomial in r bound is conjectured) and implies the existence of large constant in the actual time-complexity of the algorithms and thus make the algorithms not efficient practically. Hence an issue is to find the tight upper bound or at least lower the actual upper bound. In [26] , we show that a graph with no 3×3 grid minor has treewidth at most 7. This is tight and improves the best known upper bound which was 2942.

In [22] , we present a result concerning the relation between the path-width of a planar graph and the path-width of its topological dual. More precisely, we prove that for a 3-connected planar graph G , pw(G) $\le$ 3pw(G^*) + 2 . For 4-connected planar graphs, and more generally for Hamiltonian planar graphs, we prove a stronger bound pw(G^*) $\le$ 2pw(G) + c . The best previously known bound was obtained by Fomin and Thilikos who proved that pw(G^*) $\le$ 6pw(G) + cte . The proof is based on an algorithm which, given a fixed spanning tree of G , transforms any given decomposition of G into one of G^* . The ratio of the corresponding parameters is bounded by the maximum degree of the spanning tree. In [82] , we present a result concerning the relation between the branch-with of a graph embedded in a surface of Euler genus g and the branch-width of its topological dual. We prove that $Im12 ${{\#119835 w}{(G^*)}\#8804 6×{\#119835 w}{(G)}+2g-4}$$ for any graph G embedded in a surface of Euler genus g .

Random graphs

We studied various parameters of random graphs and random walks.

In [65] , we investigate the giant component problem in random graphs with a given degree sequence. We generalize the critical condition of Molloy and Reed [Molloy, M., and B. Reed, Random Structures Algorithms 6 (1995), 161-179], which determines the existence of a giant component in such a random graph, in order to include degree sequences with heavy tails. We show that the quantity which determines the existence of a giant component is the value of the smallest fixed point inside the interval [0, 1] of the generating function $Im13 ${F{(s)}=\#8721 _{i\#8805 1}\#948 _is^{i-1}}$$ , where $\delta$ _i is the asymptotic proportion of the total degree contained in vertices of degree i . Moreover, we show that this quantity also determines the existence of a core (i.e., the maximal subgraph of minimum degree at least 2) that has linear total degree.

In [19] , we consider the complete graph on n vertices whose edges are weighted by independent and identically distributed edge weights and build the associated minimum weight spanning tree. We show that if the random weights are all distinct, then the expected diameter of such a tree is $\upper_theta$ (n^1/3) . This settles a question of Frieze and McDiarmid [Random Structures Algorithms, 10(1-2):5–42, 1997]. The proofs are based on a precise analysis of the behaviour of random graphs around the critical point.

Given a branching random walk, let M_n be the minimum position of any member of the n th generation. In [20] , we calculate $Im14 ${\#120124 (M_n)}$$ to within O(1) and prove exponential tail bounds for $Im15 ${\#8473 (|M_n-\#120124 {(M_n)}|\gt x)}$$ , under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89-108], our results fully characterise the possible behaviour of $Im14 ${\#120124 (M_n)}$$ when the branching random walk has bounded branching and step size.

Miscellaneous

In [40] we prove the following result. Suppose that s and t are vertices of a 3-connected graph G such that G-{s, t} is not bipartite and there is no cutset X of size three in G for which some component U of G-X is disjoint from {s, t} . Then either (1) G contains an induced path P from s to t such that G-V(P) is not bipartite or (2) G can be embedded in the plane so that every odd face contains one of s or t . Furthermore, if (1) holds then we can insist that G-V(P) is connected, while if G is 5-connected then (1) must hold and P can be chosen so that G-V(P) is 2-connected.

A circuit in a simple undirected graph G = (V, E) is a sequence of vertices {v₁, v₂, ..., v_{k + 1}} such that v₁ = v_{k + 1} and {v_i, v_{i + i}} $\in$ E for i = 1, ..., k . A circuit C is said to be edge-simple if no edge of G is used twice in C . In [107] , [57] , we study the following problem: which is the largest integer k such that, given any subset of k ordered vertices of an infinite square grid, there exists an edge-simple circuit visiting the k vertices in the prescribed order? We prove that k = 10 . To this end, we first provide a counterexample implying that k<11 . To show that k $\ge$ 10 , we introduce a methodology, based on the notion of core graph, to reduce drastically the number of possible vertex configurations, and then we test each one of the resulting configurations with an ILP solver.

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