Bruce Reed

McGill University | McGill · School of Computer Science

We show that for sufficiently large $d$ and for $t\geq d+1$, there is a graph $G$ with average degree $(1-\varepsilon)\lambda t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$, where $\lambda=0.63817\dots$ is an explicitly defined constant. This generalises analogous results for complete...

We show that for sufficiently large $d$ and for $t\geq d+1$, there is a graph $G$ with average degree $(1-\varepsilon)\lambda t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$, where $\lambda=0.63817\dots$ is an explicitly defined constant. This generalises analogous results for complete...

In this paper we show how to combine two algorithmic techniques to obtain linear time algorithms for various optimization problems on graphs, and present a subroutine which will be useful in doing so. The first technique is iterative shrinking. In the first phase of an iterative shrinking algorithm, we construct a sequence of graphs of decreasing s...

Ramsey's theorem says that for every clique H1 and for every graph H2 with no edges, all graphs containing neither of H1, H2 as induced subgraphs have bounded order. What if, instead, we exclude a graph H1 with a vertex whose deletion gives a clique, and the complement H2 of another such graph? This no longer implies bounded order, but it implies t...

We prove a conjecture of Ohba which says that every graph $G$ on at most $2\chi(G)+1$ vertices satisfies $\chi_\ell(G)=\chi(G)$.

We consider the following problem. For every positive integer k there is a smallest integer f(k)f(k) such that for any two vertices s and t in a non-bipartite f(k)f(k)-connected graph G, there is an s–t path P in G with specified parity such that G−V(P)G−V(P) is k-connected. This conjecture is a variant of the well-known conjecture of Lovász with t...

Let D be a digraph. The chromatic number chi(D) of D is the smallest number of colors needed to color the vertices of D such that every color class induces an acyclic subdigraph. The girth of D is the length of a shortest directed cycle, or infinity if D is acyclic. Let G(k, n) be the maximum possible girth of a digraph on n vertices with chi(D) >...

Let f(k) be the smallest integer such that every f(k)-chromatic digraph contains every oriented tree of order k. Burr proved f(k)≤(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)≤k 2 /2-k/2+1 and that every ant...

The second author's $\omega$, $\Delta$, $\chi$ conjecture proposes that every graph satisties $\chi \leq \lceil \frac 12 (\Delta+1+\omega)\rceil$. In this paper we prove that the conjecture holds for all claw-free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way we discuss a stronger local conjecture, and pro...

In 1998 the second author proved that there is an $\epsilon>0$ such that every graph satisfies $\chi \leq \lceil (1-\epsilon)(\Delta+1)+\epsilon\omega\rceil$. The first author recently proved that any graph satisfying $\omega > \frac 23(\Delta+1)$ contains a stable set intersecting every maximum clique. In this note we exploit the latter result to...

A totally odd K4-subdivision is a subdivision of K4 where each subdivided edge has odd length. The recognition of a totally odd K4-subdivision plays an important role in both graph theory and combinatorial optimization. Sewell and Trotter [53], Zang [63] and Thomassen [60] independently conjectured the existence of a polynomial time recognition alg...

We consider the following well-known problem, which is called the disjoint paths problem. For a given graph G and a set of k pairs of terminals in G, the objective is to find k vertex-disjoint paths connecting given pairs of terminals or to conclude that such paths do not exist. We present an O(n2) time algorithm for this problem for fixed k. This...

Let g(t) be the minimum number such that every graph G with average degree d(G) \geq g(t) contains a K_{t}-minor. Such a function is known to exist, as originally shown by Mader. Kostochka and Thomason independently proved that g(t) \in \Theta(t*sqrt{log t}). This article shows that for all fixed \epsilon > 0 and fixed sufficiently large t \geq t(\...

An $L(p,1)$-labeling of a graph is a function $f$ from the vertex set to the positive integers such that $|f(x)-f(y)|\geqslant p$ if dist$(x,y)=1$ and $|f(x)-f(y)|\geqslant 1$ if dist$(x,y)=2$, where dist$(x,y)$ is the distance between the two vertices $x$ and $y$ in the graph. The span of an $L(p,1)$-labeling $f$ is the difference between the larg...

We generalize the seminal Graph Minor algorithm of Robertson and Seymour to the parity version. We give polynomial time algorithms for the following problems: 1) the parity H-minor (Odd K k-minor) containment problem, 2) the disjoint paths problem with k terminals and the parity condition for each path, as well as several other related problems. We...

A class A of labelled graphs is bridge-addable if for all graphs G in A and all vertices u and v in distinct connected components of G, the graph obtained by adding an edge between u and u is also in A; the class A is monotone if for all G in A and all subgraphs H of G, H is also in A. We show that for any bridge-addable, monotone class A whose ele...

The clique cover number of a graph G is the minimum number of cliques required to cover the edges of graph G. In this paper we consider the random graph G(n,p), for p constant. We prove that with probability 1-o(1), the clique number of G(n,p) is Theta(n^2/\log^2n).

P. Erdős and A. Hajnal [“Ramsey-type theorems,” Discrete Appl. Math. 25, No. -2, 37–52 (1989; Zbl 0715.05052)] conjectured that, for every graph H, there exists a constant ε(H)>0 such that every H-free graph G (that is, not containing H as an induced subgraph) must contain a clique or an independent set of size at least |G| ε(H) . We prove that the...

As proved by J. Kahn [J. Comb. Theory, Ser. B 68, No. 2, 233–254 (1996; Zbl 0861.05026)], the chromatic number and fractional chromatic number of a line graph agree asymptotically.That is, for any line graph G, we have χ(G)≤(1+o(1))χ f (G). We extend this result to quasi-line graphs, an important subclass of claw-free graphs. Furthermore, we prove...

Let H be a graph. If G is an n-vertex simple graph that does not contain H as a minor, what is the maximum number of edges that G can have? This is at most linear in n, but the exact expression is known only for very few graphs H. For instance, when H is a complete graph Kt, the “natural” conjecture, , is true only for t⩽7 and wildly false for larg...

The term Probabilistic Method refers to the proof of deterministic statements using probabilistic tools. The method has been successfully applied to a number of problems in the field of graph colouring. We survey some of the results thereby obtained. The talk is intended to be accessible and short on details. We will first define graph colouring, e...

It is shown that for each t, there is a separator of size O(t√n) in any n-vertex graph G with no Kt-minor. This settles a conjecture of Alon, Seymour and Thomas (J. Amer. Math. Soc, 1990 and STOC'90), and generalizes a result of Djidjev (1981), and Gilbert, Hutchinson and Tarjan (J. Algorithm, 1984), independently, who proved that every graph with...

We consider the following problem, which is called the odd cycle packing problem. Input: A graph $G$ with n vertices and m edges, and an integer k. Output: k vertex disjoint odd cycles. We also consider the edge disjoint case, and the node- and arc-disjoint directed case. This problem is known to be NP-hard, even for planar graphs, if k is part of...

An i-triangulated graph is a graph in which every odd cycle has two non-crossing chords; i-triangulated graphs form a subfamily of perfect graphs. A slightly more general family of perfect graphs are clique-separable graphs. A graph is clique-separable precisely if every induced subgraph either has a clique cutset, or is a complete multipartite gra...

We correct an error that appears in [M. Molloy, B. Reed, Asymptotically optimal frugal colouring, J. Combin. Theory Ser. B 100 (2) (2010) 226–246, this issue].

We prove that every graph with maximum degree Δ can be properly (Δ+1)-coloured so that no colour appears more than times in the neighbourhood of any vertex. This is best possible up to the constant multiple in the O(−) term.

We consider the following problem, which is called the odd cycles transversal problem. Input: A graph G and an integer k. Output: A vertex set X ∈ V(G) with |X| ≤ k such that G -- X is bipartite. We present an O(mα(m, n)) time algorithm for this problem for any fixed k, where n, m are the number of vertices and the number of edges, respectively, an...

A totally odd K4-subdivision is a subdivision of K4 where each subdivided edge has odd length. The recognition of a totally odd K4-subdivision plays an important role in both graph theory and combinatorial optimization. Sewell and Trotter [53], Zang [63] and Thomassen [60] independently conjectured the existence of a polynomial time recognition alg...

We prove that every graph with maximum degreecan be properly (� + 1)- coloured so that no colour appears more than O(log �/log log �) times in the neigh- bourhood of any vertex. This is best possible up to the constant multiple in the O(−) term.

For a simple graph of maximum degree Δ, the complexity of computing the fractional total chromatic number is unknown. Trivially it is at least Δ+1. Kilakos and Reed proved that it is at most Δ+2. In this paper, we strengthen this by characterizing exactly those simple graphs with fractional total chromatic number Δ+2. This yields a simple linear-ti...

Let G be an n-vertex m-edge graph with weighted vertices. A pair of vertex sets A, B ⊆ V(G) is a 2/3-separation of order |A ∩ B| if A ∪ B = V(G), there is no edge between A − B and B − A, and both A − B and B − A have weight at most 2/3 the total weight of G. Let ℓ ∈ Z+ be fixed. Alon et al. [1990] presented an algorithm th...

For any c>1, we describe a linear time algorithm for fractionally edge colouring simple graphs with maximum degree at least |V|/c.

In this contribution, 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, A critical point for random graphs with given degree sequence, Random Structures Algorithms 6 (1995), 161–179], which determines the existence of a giant co...

We show that graphs with no minor isomorphic to the 3×3 grid have tree-width at most 7.

The famous Hadwiger's conjecture asserts that every graph with no Kt-minor is (t-1)-colorable. The case t=5 is known to be equivalent to the Four Color Theorem by Wagner, and the case t=6 is settled by Robertson, Seymour and Thomas. So far the cases t ≥ 7 are wide open. In this paper, we prove the following two theorems: There is an O(n2) algorithm...

A graph G is k-linked if G has at least 2k vertices, and for any 2k vertices x 1,x 2, …, x k ,y 1,y 2, …, y k , G contains k pairwise disjoint paths P 1, …, P k such that P i joins x i and y i for i = 1,2, …, k. We say that G is parity-k-linked if G is k-linked and, in addition, the paths P 1, …, P k can be chosen such that the parities of th...

In this paper 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 embe...

A Kl-expansion consists of l 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 l, if a graph contains no odd Kl-expansion then its chromatic number is...

We consider the following problem, which is called the half integral parity disjoint paths packing problem. Input: A graph G, k pair of vertices (s1,t1),(s2,t2),...,(sk,tk) in G (which are some- times called terminals), and a parity li for each i with 1 i k, where li = 0 or 1. Output : Paths P1,...,Pk in G such that Pi joins si and ti for i = 1,2,....

A hole in a graph is an induced subgraph which is a cycle of length at least four. A hole is called even if it has an even number of vertices. An even-hole-free graph is a graph with no even holes. A vertex of a graph is bisimplicial if the set of its neighbours is the union of two cliques. In this paper we prove that every even-hole-free graph has...

A quasi-line graph is a graph in which the neighborhood of any vertex can be covered by two cliques; every line graph is a quasi-line graph. Reed conjectured that for any graph G, [Reed, J Graph Theory 27 (1998), 177–212]. We prove that the conjecture holds if G is a quasi-line graph, extending a result of King et al. who proved the conjecture for...

We begin by sketching the development of the classical ballot theorem as it first appeared in the Comptes Rendus de 1’Academie des Sciences. The statement that is fairly called the first Ballot Theorem was due to Bertrand: Theorem 1 ([8]). We suppose that two candidates have been submitted to a vote in which the number of voters is μ. Candidate A o...

For every fixed surface S, orientable or non-orientable, and a given graph G, Mohar (STOC'96 and Siam J. Discrete Math. (1999)) described a linear time algorithm which yields either an embedding of G in S or a minor of G which is not embeddable in S and is minimal with this property. That algorithm, however, needs a lot of lemmas which spanned six...

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $\ell\times\ell$ grid minor is exponential in $\ell$. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A \emph{grid-like-minor of order} $...

We prove there exists a function f(k) such that for every f(k)-connected graph G and for every edge e 2 E(G), there exists an induced cycle C containing e such that G E(C) is k-connected. This proves a weakening of a conjecture of Lovasz due to Kriesell.

In this article we present a study of the mixing time of a random walk on the largest component of a supercritical random graph, also known as the giant component. We identify local obstructions that slow down the random walk, when the average degree d is at most O(), proving that the mixing time in this case is Θ((n/d)2) asymptotically almost sure...

In 1977, Wegner conjectured that the chromatic number of the square of every planar graph $G$ with maximum degree $\Delta\ge8$ is at most $\bigl\lfloor\frac32\Delta\bigr\rfloor+1$. We show that it is at most $\frac32 \Delta (1+o(1))$ (where the $o(1)$ is as $\Delta\to+\infty$), and indeed that this is true for the list chromatic number and for more...

Let the random graph Rn be drawn uniformly at random from the set of all simple planar graphs on n labelled vertices. We see that with high probability the maximum degree of Rn is Θ(ln n). We consider also the maximum size of a face and the maximum increase in the number of components on deleting a vertex. These results extend to graphs embeddable...

In 1977, Wegner conjectured that the chromatic number of the square of every planar graph G with maximum degree Δ⩾8 is at most . We show that it is at most , and indeed this is true for the list chromatic number.

We present a linear time algorithm which determines whether an input graph contains K 5 as a minor and outputs a K 5-model if the input graph contains one. If the input graph has no K 5-minor then the algorithm constructs a tree decomposition such that each node of the tree corresponds to a planar graph or a graph with eight vertices. Such a decomp...

In this paper, we present new structural results about the existence of a subgraph where the degrees of the vertices are pre-specified. Further, we use these results to prove a 16-edge-weighting version of a conjecture by Karoński, Łuczak and Thomason, an asymptotic 2-edge-weighting version of the same conjecture, and a 78 version of Louigi's Conje...

We discuss some new and old results about skew partitions in perfect graphs.

We study the fractional total chromatic number of Gn,p as p varies from 0 to 1. We also present an algorithm that computes the fractional total chromatic number of a random graph in polynomial expected time.

We prove an analogue of the classical ballot theorem that holds for any random walk in the range of attraction of the normal distribution. Our result is best possible: we exhibit examples demonstrating that if any of our hypotheses are removed, our conclusions may no longer hold.

In this paper we prove a result about vertex list colourings which in particular shows that a conjecture of the second author (1999, Journal of Graph Theory 31, 149-153) is true for triangle free graphs of large maximum degree. There exists a constant K such that the following holds: Given a graph G and a list assignment L to vertices of G, assigni...

Gerards and Seymour (see (T.R. Jensen, B. Toft, Graph Coloring Problems, Wiley-Interscience, 1995), page 115) conjectured that if a graph has no odd complete minor of order p, then it is (p 1)-colorable. This is an analogue of the well known conjecture of Hadwiger, and in fact, this would immediately imply Hadwiger's conjecture. The current best kn...

An L(2,1)-labelling of a graph is a function / from the vertex set to the positive integers such that \f(x) - f(y)\ ≥ 2 if dist(x,y) = 1 and \f(x) - f(y)\ ≥ 1 if dist(x,y) = 2, where dist(x, y) is the distance between the two vertices × and y in the graph G. The span of an L(2,1)-labelling f is the difference between the largest and the smallest la...

A graph $G$ is strict quasi parity (SQP) if every induced subgraph of $G$ that is not a clique contains a pair of vertices with no odd chordless path between them (an even pair). Hougardy conjectured that the minimal forbidden subgraphs for the class of SQP graphs are the odd chordless cycles, the complements of odd or even chordless cycles, and so...

Abstract For each constant k, we present a linear time algorithm that, given a planar graph G, either finds a minimum odd cycle vertex transversal in G or guarantees that there is no transversal of size at most k. © 2007 Elsevier B.V. All rights reserved. Keywords: Planar graph; Odd cycle; Linear time algorithm

Given a branching random walk, let $M_n$ be the minimum position of any member of the $n$th generation. We calculate $\mathbf{E}M_n$ to within O(1) and prove exponential tail bounds for $\mathbf{P}\{|M_n-\mathbf{E}M_n|>x\}$, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw....

The Clique-width of a graph is an invariant which measures the complexity of the graph structures. A graph of bounded tree-width is also of bounded Clique-width (but not the converse). For graphs G of bounded , given the bounded width decomposition of G, every optimization, enumeration or evaluation problem that can be defined by a Monadic Second O...

2 m , where ( G) and !(G) are the maximum degree and clique number of G, respectively. In this paper we prove that this bound holds if G is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph G and produces a colouring that achieves our bound.

In 1977, Wegner conjectured that the chromatic number of the square of every planar graph G with maximum degree Δ⩾8 is at most View the MathML source. We show that it is at most View the MathML source, and indeed this is true for the list chromatic number.

We show that for every fixed k, there is a linear time algorithm that decides whether or not a given graph has crossing number at most k, and if this is the case, computes a drawing of the graph in the plane with at most k crossings. This answers the question posed by Grohe (STOC'01 and JCSS 2004). Our algorithm can be viewed as a generalization of...

A skew partition as defined by Chvátal is a partition of the vertex set of a graph into four nonempty parts A, B, C, D such that there are all possible edges between A and B, and no edges between C and D. We present a polynomial-time algorithm for testing whether a graph admits a skew partition. Our algorithm solves the more general list skew parti...

We prove a new bound on the mixing time of a Markov chain by considering the conductance of its connected subsets.

We show that, for every l, the family $ \mathcal{F}_{l} $ \mathcal{F}_{l} of circuits of length at least l satisfies the Erdős–Pósa property, with f(k)=13l(k−1)(k−2)+(2l+3)(k−1), thereby sharpening a result of C. Thomassen. We obtain as a corollary that graphs without k disjoint circuits of length l or more have tree-width O(lk2).

In this paper we present a study of the mixing time of a random walk on the largest component of a supercritical random graph, also known as the giant component. We identify local obstructions that slow down the random walk, when the average degree d is at most (ln n lnln n)^{1/2}, proving that the mixing time in this case is O((ln n/d)^2) asymptot...

A weighting w of the edges of a graph G induces a colouring of the vertices of G where the colour of vertex v, denoted c v , is \( {\sum\nolimits_{e \mathrel\backepsilon v} {w{\left( e \right)}} } \). We show that the edges of every graph that does not contain a component isomorphic to K 2 can be weighted from the set {1, . . . ,30} such that in th...

We study the ratio between the minimum size of an odd cycle vertex transversal and the maximum size of a collection of vertex-disjoint odd cycles in a planar graph. We show that this ratio is at most 10. For the corresponding edge version of this problem, Král and Voss recently proved that this ratio is at most 2; we also give a short proof of thei...

This is the story of a man named Vašek.

Using the symmetric form of the Lovász Local Lemma, one can conclude that a k-uniform hypergraph \(\mathcal{H}\) admits a proper 2-colouring if the maximum degree (denoted by Δ) of \(\mathcal{H}\) is at most \(\frac{2^k}{8k}\) independently of the size of the hypergraph. However, this argument does not give us an algorithm to find a proper 2-colour...

Chvátal defined a skew partition of a graph G to be a partition of its vertex set into two non-empty parts A and B such that A induces a disconnected subgraph of G and B induces a disconnected subgraph of [`(G)]\overline{G} . Skew partitions are important in the characterization of perfect graphs. De Figuereido et al. presented a polynomial time a...

We prove that connected cubic graphs of order n and girth g have domination number at most 0.32127 n+O(\fracng)n+O\left(\frac{n}{g}\right) .

We see that the entropy method yields strong concentration results for general self-bounding functions of independent random variables. These give an improvement of a concentration result of Talagrand much used in discrete mathematics. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006

International audience Let $X_1,\ldots,X_{n\choose 2}$ be independent identically distributed weights for the edges of $K_n$. If $X_i \neq X_j$ for$ i \neq j$, then there exists a unique minimum weight spanning tree $T$ of $K_n$ with these edge weights. We show that the expected diameter of $T$ is $Θ (n^{1/3})$. This settles a question of [Frieze97...

We consider the lower bound for building a heap in the worst case and the upper bound in the average case. We will prove that the supposedly fastest algorithm in the average case[2] does not attain its claimed bound and indeed is slower than that in [6]. We will then prove that the adversarial argument for the claimed best lower bound in the worst...

The chromatic index c e(G )o fa nu ndirected graph G is the minimum number of matchings needed to partition its edge set. Let Δ(G )d enote the maxi- mum verte xd egree of G ,a nd let G denote the complement of G.J ensen and Toft conjectured that for a graph G with an e ve nn umber of v ertices, either c e(G) =Δ ( G )o r c e ( G) =Δ (G). W ep rove a...

A partition of the edges of a graph G into sets {S1,…,Sk} defines a multiset Xv for each vertex v where the multiplicity of i in Xv is the number of edges incident to v in Si. We show that the edges of every graph can be partitioned into 4 sets such that the resultant multisets give a vertex colouring of G. In other words, for every edge (u,v) of G...

We consider the class A 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 show that the coloring algorithm found by the second and fourth author can be implemented in time O(n^2m) for any graph in A with n vertices and m edges, thereby improvin...

We study the ratio between the minimum size of an odd cycle vertex transversal and the maximum size of a collection of vertex-disjoint odd cycles in a planar graph. We show that this ratio is at most10. For the corresponding edge version of this problem, Král and Voss [7] recently proved that this ratio is at most 2; we also give a short proof of t...

The recently announced Strong Perfect Graph Theorem states that the class of perfect graphs coincides with the class of graphs containing no induced odd cycle of length at least 5 or the complement of such a cycle. A graph in this second class is called Berge. A bull is a graph with five vertices x, a, b, c, d and five edges xa, xb, ab, ad, bc. A g...

Every graph G of maximum degree Δ is (Δ + 1)-colourable and a classical theorem of Brooks states that G is not Δ-colourable iff G has a (Δ + 1)-clique or Δ = 2 and G has an odd cycle. Reed extended Brooks' Theorem by showing that if Δ(G) ≥ 1014 then G is not (Δ - 1)-colourable iff G contains a Δ-clique. We extend Reed's characterization of (Δ - 1)-...

Let ${\cal F}$ be a family of graphs. Two graphs G1 = (V1,E1), G2=(V2,E2) are said to have the same ${\cal F}$-structure if there is a bijection $f: V_1 \rightarrow V_2$ such that a subset S induces a graph belonging to ${\cal F}$ in G1 if and only if its image f(S) induces a graph belonging to ${\cal F}$ in G2. We characterize those graphs which h...

For each constant k, we present a linear time algorithm that, given a planar graph G, either finds a minimum odd cycle vertex transversal in G or guarantees that there is no transversal of size at most k.

International audience Let $G$ be an $n$-vertex $m$-edge graph with weighted vertices. A pair of vertex sets $A,B \subseteq V(G)$ is a $\frac{2}{3} - \textit{separation}$ of $\textit{order}$ $|A \cap B|$ if $A \cup B = V(G)$, there is no edge between $A \backslash B$ and $B \backslash A$, and both $A \backslash B$ and $B \backslash A$ have weight a...

A skew partition is a partition of the vertex set of a graph into four nonempty parts A,B,C,D such that there are all possible edges between A and B, and no edges between C and D. A stable skew partition is a skew partition where A induces a stable set of the graph. We show that determining if a graph permits a stable skew partition is NP-complete....

We prove that Hadwiger’s conjecture holds for line graphs. Equivalently, we show that for every loopless graph G (possibly with parallel edges) and every integer k≥0, either G is k-edge-colourable, or there are k+1 connected subgraphs A1,…,Ak+1 of G, each with at least one edge, such that E(Ai∩Aj)=∅ and V(Ai∩Aj)≠∅ for 1≤i

We present an O(mn) algorithm to determine whether a graph G with m edges and n vertices has an odd cycle transversal of order at most k, for any fixed k. We also obtain an algorithm that determines, in the same time, whether a graph has a half integral packing of odd cycles of weight k.

This article proves the conjecture of Thomas that, for every graph G, there is an integer k such that every graph with no minor isomorphic to G has a 2-coloring of either its vertices or its edges where each color induces a graph of tree-width at most k. Some generalizations are also proved.