K-colourable Graphs Research Articles

Reconfiguration of vertex colouring and forbidden induced subgraphs

The reconfiguration graph of the k-colourings, denoted Rk(G), is the graph whose vertices are the k-colourings of G and two colourings are adjacent in Rk(G) if they differ in colour on exactly one vertex. In this paper, we investigate the connectivity and diameter of Rk+1(G) for a k-colourable graph G restricted by forbidden induced subgraphs. We show that Rk+1(G) is connected for every k-colourable H-free graph G if and only if H is an induced subgraph of P4 or P3+P1. We also start an investigation into this problem for classes of graphs defined by two forbidden induced subgraphs. We show that if G is a k-colourable (2K2, C4)-free graph, then Rk+1(G) is connected with diameter at most 4n. Furthermore, we show that Rk+1(G) is connected for every k-colourable (P5, C4)-free graph G.

Mixing colourings in 2K2-free graphs

The reconfiguration graph for the k-colourings of a graph G, denoted Rk(G), is the graph whose vertices are the k-colourings of G and two colourings are joined by an edge if they differ in colour on exactly one vertex. For any k-colourable P4-free graph G, Bonamy and Bousquet proved that Rk+1(G) is connected. In this short note, we complete the classification of the connectedness of Rk+1(G) for a k-colourable graph G excluding a fixed path, by constructing a 7-chromatic 2K2-free (and hence P5-free) graph admitting a frozen 8-colouring. This settles a question of the second author.

A spanning bandwidth theorem in random graphs

AbstractThe bandwidth theorem of Böttcher, Schacht and Taraz states that any n-vertex graph G with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). Recently, a subset of the authors proved a random graph analogue of this statement: for $p\gg \big(\tfrac{\log n}{n}\big)^{1/\Delta}$ a.a.s. each spanning subgraph G of G(n,p) with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all n-vertex k-colourable graphs H with maximum degree $\Delta$ , bandwidth o(n), and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper, we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in G contain many copies of $K_\Delta$ then we can drop the restriction on H that $Cp^{-2}$ vertices should not be in triangles.

A construction of uniquely colourable graphs with equal colour class sizes

A uniquelyk-colourable graph is a graph with exactly one partition of the vertex set into at most k colour classes. Here, we investigate some constructions of uniquely k-colourable graphs and give a construction of Kk-free uniquely k-colourable graphs with equal colour class sizes.

A note on perfect Lucky k-colourable graphs

A note on perfect Lucky k-colourable graphs

Join colourings of chordal graphs

We consider certain partition problems typified by the following two questions. When is a given graph the join of two k-colourable graphs? When is a given graph the join of a k-colourable graph and a graph whose complement is k-colourable? We focus on the class of chordal graphs, and employ the language of matrix partitions. Our emphasis is on forbidden induced subgraph characterizations. We describe all chordal minimal obstructions to the existence of such partitions; they are surprisingly small and regular. We also give another characterization which yields a more efficient recognition algorithm. By contrast, most of these partition problems are NP-complete if chordality is not assumed.

The property of [formula omitted]-colourable graphs is uniquely decomposable

An additive hereditary graph property is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. If P1,…,Pn are graph properties, then a (P1,…,Pn)-decomposition of a graph G is a partition E1,…,En of E(G) such that G[Ei], the subgraph of G induced by Ei, is in Pi, for i=1,…,n. The sum of the properties P1,…,Pn is the property P1⊕⋯⊕Pn={G∈I:G has a (P1,…,Pn)-decomposition}. A property P is said to be decomposable if there exist non-trivial additive hereditary properties P1 and P2 such that P=P1⊕P2. A property is uniquely decomposable if, apart from the order of the factors, it can be written as a sum of indecomposable properties in only one way. We show that not all properties are uniquely decomposable; however, the property of k-colourable graphs Ok is a uniquely decomposable property.

Universal H-Colourable Graphs

Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: for given m and n with m < n, m is adjacent to n if n has a 1 in the mth position of its binary expansion. It is well known that R is a universal graph in the set $${\mathcal{I}_c}$$ of all countable graphs (since every graph in $${\mathcal{I}_c}$$ is isomorphic to an induced subgraph of R) and that it is a homogeneous graph (since every isomorphism between two finite induced subgraphs of R extends to an automorphism of R). In this paper we construct a graph U(H) which is H-universal in ? H c , the induced-hereditary hom-property of H-colourable graphs consisting of all (countable) graphs which have a homomorphism into a given (countable) graph H. If H is the (finite) complete graph K k , then ? H c is the property of k-colourable graphs. The universal graph U(H) is characterised by showing that it is, up to isomorphism, the unique denumerable, H-universal graph in ? H c which is H-homogeneous in ? H c . The graphs H for which $${U(H) \cong R}$$ are also characterised. With small changes to the definitions, our results translate effortlessly to hold for digraphs too. Another slight adaptation of our work yields related results for (k, l)-split graphs.

Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs

A k-colouring of a graph G=(V,E) is a mapping c:V→{1,2,…,k} such that c(u)≠c(v) whenever uv is an edge. The reconfiguration graph of the k-colourings of G contains as its vertex set the k-colourings of G, and two colourings are joined by an edge if they differ in colour on just one vertex of G. We introduce a class of k-colourable graphs, which we call k-colour-dense graphs. We show that for each k-colour-dense graph G, the reconfiguration graph of the ℓ-colourings of G is connected and has diameter O(|V|2), for all ℓ≥k+1. We show that this graph class contains the k-colourable chordal graphs and that it contains all chordal bipartite graphs when k=2. Moreover, we prove that for each k≥2 there is a k-colourable chordal graph G whose reconfiguration graph of the (k+1)-colourings has diameter Θ(|V|2).

No-Three-in-Line-in-3D

The no-three-in-line problem, introduced by Dudeney in 1917, asks for the maximum number of points in the n × n grid with no three points collinear. Erdos proved that the answer is Θ(n). We consider the analogous problem in three dimensions, and prove that the maximum number of points in the n × n × n grid with no three collinear is Θ(n2). This result is generalised by the notion of a 3D drawing of a graph. Here each vertex is represented by a distinct gridpoint in ${\Bbb Z}^3$, such that the line-segment representing each edge does not intersect any vertex, except for its own endpoints. Note that edges may cross. A 3D drawing of a complete graph Kn is nothing more than a set of n gridpoints with no three collinear. A slight generalisation of our first result is that the minimum volume for a 3D drawing of Kn is Θ(n3/2). This compares favourably with Θ(n3) when edges are not allowed to cross. Generalising the construction for Kn, we prove that every k-colourable graph on n vertices has a 3D drawing with ${\cal O}{n\sqrt{k}}$ volume, which is optimal for the k-partite Turan graph.

Colouring Semirandom Graphs

We study semirandom k-colourable graphs made up as follows. Partition the vertex set V = {1, . . ., n} randomly into k classes V1, . . ., Vk of equal size and include each Vi–Vj-edge with probability p independently (1 ≤ i )klnn, C0k2} for a certain constant C0>0 and an arbitrarily small but constant >0, an optimal colouring of G*n,p,k can be found in polynomial time with high probability. Furthermore, if np ≥ C0max{klnn, k2}, a k-colouring of G*n,p,k can be computed in polynomial expected time. Moreover, an optimal colouring of G*n,p,k can be computed in expected polynomial time if k ≤ ln1/3n and np ≥ C0klnn. By contrast, it is NP-hard to k-colour G*n,p,k With high probability if $np\leq (\frac12-\varepsilon)k\ln(n/k)$ .

On Approximate Graph Colouring and MAX-k-CUT Algorithms Based on the θ-Function

The problem of colouring a k-colourable graph is well-known to be NP-complete, for k ≥ 3. The MAX-k-CUT approach to approximate k-colouring is to assign k colours to all of the vertices in polynomial time such that the fraction of `defect edges' (with endpoints of the same colour) is provably small. The best known approximation was obtained by Frieze and Jerrum (1997), using a semidefinite programming (SDP) relaxation which is related to the Lovász θ-function. In a related work, Karger et al. (1998) devised approximation algorithms for colouring k-colourable graphs exactly in polynomial time with as few colours as possible. They also used an SDP relaxation related to the θ-function. In this paper we further explore semidefinite programming relaxations where graph colouring is viewed as a satisfiability problem, as considered in De Klerk et al. (2000). We first show that the approximation to the chromatic number suggested in De Klerk et al. (2000) is bounded from above by the Lovász θ-function. The underlying semidefinite programming relaxation in De Klerk et al. (2000) involves a lifting of the approximation space, which in turn suggests a provably good MAX-k-CUT algorithm. We show that of our algorithm is closely related to that of Frieze and Jerrum; thus we can sharpen their approximation guarantees for MAX-k-CUT for small fixed values of k. For example, if k = 3 we can improve their bound from 0.832718 to 0.836008, and for k = 4 from 0.850301 to 0.857487. We also give a new asymptotic analysis of the Frieze-Jerrum rounding scheme, that provides a unifying proof of the main results of both Frieze and Jerrum (1997) and Karger et al. (1998) for k ≫ 0.

Grid drawings of k-colourable graphs

It is proved that every k-colourable graph on n vertices has a grid drawing with O ( k n ) area, and that this bound is best possible. This result can be viewed as a generalisation of the no-three-in-line problem. A further area bound is established that includes the aspect ratio as a parameter.

Locally bounded hereditary subclasses of k-colourable graphs

A vertex colouring C 1, C 2,…, C k of a graph G is called l- bounded ( l⩾0) if | C i⧹N(u)|⩽l for all i=1,2,…, k and for every vertex u∈VG⧹ C i ; here N( u) is the neighbourhood of u. Let C( k, l) be the class of all graphs having an l-bounded k-colouring. We show that for all k and l the class C( k, l) can be described in terms of forbidden induced subgraphs. This result implies the existence of polynomial time algorithms recognizing C( k, l). We also find the minimal set of forbidden induced subgraphs for the class C(3,1).

Algorithms for Colouring Random k-colourable Graphs

The k-colouring problem is to colour a given k-colourable graph with k colours. This problem is known to be NP-hard even for fixed k [ges ] 3. The best known polynomial time approximation algorithms require nδ (for a positive constant δ depending on k) colours to colour an arbitrary k-colourable n-vertex graph. The situation is entirely different if we look at the average performance of an algorithm rather than its worst-case performance. It is well known that a k-colourable graph drawn from certain classes of distributions can be k-coloured almost surely in polynomial time.In this paper, we present further results in this direction. We consider k-colourable graphs drawn from the random model in which each allowed edge is chosen independently with probability p(n) after initially partitioning the vertex set into k colour classes. We present polynomial time algorithms of two different types. The first type of algorithm always runs in polynomial time and succeeds almost surely. Algorithms of this type have been proposed before, but our algorithms have provably exponentially small failure probabilities. The second type of algorithm always succeeds and has polynomial running time on average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work as long as p(n) [ges ] n−1+ε where ε is a constant greater than 1/4.

The existence of uniquely − G colourable graphs

Given graphs F and G and a nonnegative integer k, a function π : V( F) → 1, …, k is a − G k-colouring of F if no induced copy of is monochromatic; F is − G k-chromatic if F has a − G k-colouring but no − G ( k − 1)-colouring. Further, we say F is uniquely − G k-colourable if F is − G k-chromatic and, up to a permutation of colours, it has only one − G k-colouring. Such notions are extensions of the well-known corresponding definitions from chromatic theory. It was conjectured that for all graphs G of order at least two and all positive integers k there exist uniquely − G k-colourable graphs. We prove the conjecture and show that, in fact, in all cases infinitely many such graphs exist.

ON UNIQUELY -G k-COLOURABLE GRAPHS

Abstract Given graphs F and G and a nonnegative integer k, a map Π: V(F) → + {lm …, k} is a -G k-colouring of F if the subgraphs induced by each colour class do not contain G as an induced subgraph; F is -G k-chromatic if F has a -G k-colouring but no -G (k—1)-colouring. Further, we say F is uniquely -G k-colourable if and only if F is -G k-chromatic and, up to a permutation of colours, it has only one -G k-colouring. Such notions are extensions of the well known corresponding definitions from chromatic theory. In a previous paper (J. Graph. Th. 11 (1987), 87–99), the authors conjectured that for all graphs G of order at least two and all nonnegative integers k there exist uniquely -G k-colourable graphs. We show here that the conjecture holds whenever G or its complement is 2-connected.