Conjecture 2 colour theorem

2008

Constraint satisfaction problems have enjoyed much attention since the early seventies, and in the last decade have become also a focus of attention amongst theoreticians. Graph colourings are a special class of constraint satisfaction problems; they offer a microcosm of many of the considerations that occur in constraint satisfaction. From the point of view of theory, they are well known to exhibit a dichotomy of complexity -the k-colouring problem is polynomial time solvable when k ≤ 2, and NP-complete when k ≥ 3. Similar dichotomy has been proved for the class of graph homomorphism problems, which are intermediate problems between graph colouring and constraint * Supported by NSERC. † Partially supported by grant 1M0021620808 of the Czech Ministry of Educations and by AEOLUS.

2015

A rigidity theory is developed for bar-joint frameworks in $\mathbb{R}^{d+1}$ whose vertices are constrained to lie on concentric $d$-spheres with independently variable radii. In particular, combinatorial characterisations are established for the rigidity of generic frameworks for $d=1$ with an arbitrary number of independently variable radii, and for $d=2$ with at most two variable radii. This includes a characterisation of the rigidity or flexibility of uniformly expanding spherical frameworks in $\mathbb{R}^{3}$. Due to the equivalence of the generic rigidity between Euclidean space and spherical space, these results interpolate between rigidity in 1D and 2D and to some extent between rigidity in 2D and 3D. Symmetry-adapted counts for the detection of symmetry-induced continuous flexibility in frameworks on spheres with variable radii are also provided.

Discrete Mathematics, 2000

We investigate the complexity of the H -colouring problem restricted to graphs of bounded degree. The H -colouring problem is a generalization of the standard c-colouring problem, whose restriction to bounded degree graphs remains NP-complete, as long as c is smaller than the degree bound (otherwise we can use the theorem of Brooks to obtain a polynomial time algorithm). For H -colouring of bounded degree graphs, while it is also the case that most problems are NP-complete, we point out that, surprisingly, there exist polynomial algorithms for several of these restricted colouring problems. Our main objective is to propose a conjecture about the complexity of certain cases of the problem. The conjecture states that for graphs of chromatic number three, all situations which are not solvable by the colouring algorithm inherent in the theorem of Brooks are NP-complete. We motivate the conjecture by proving several supporting results.

Discrete Mathematics, 2005

A bound quiver is a digraph together with a collection of specified directed walks. Given an undirected graph G, and a collection of walks I , the bound quiver recognition problem asks: Is there an orientation of G such that each walk in I is directed? We present a polynomial time algorithm for this problem, and a structural characterization of which trees can be oriented as bound quivers.

University of Malta, 1999

A graph which is isomorphic to its complement is said to be a self-complementary graph, or sc-graph for short. These graphs have a high degree of structure, and yet they are far from trivial. Suffice to say that the problem of recognising self-complementary graphs, and the problem of checking two sc-graphs for isomorphism, are both equivalent to the graph isomorphism problem.

2002

A technique is described that constructs a 4-colouring of a planar triangulation in quadratic time. The method is based on iterating Kempe's technique. The heuristic gives rise to an interesting family of graphs which cause the algorithm to cycle. The structure of these graphs is described. A modified algorithm that appears always to work is presented. These techniques may lead to a proof of the 4-Colour Theorem which does not require a computer to construct and colour irreducible configurations.

Computing Research Repository, 2011

We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems, surjective homomorphism problems seem to be harder to classify and we examine especially three concrete problems that have arisen from the literature, two of which remain of

Graphs and Combinatorics, 2015

Let G be a plane graph. A vertex-colouring ϕ of G is called facial non-repetitive if for no sequence r 1 r 2 . . . r 2n , n ≥ 1, of consecutive vertex colours of any facial path it holds r i = r n+i for all i = 1, 2, . . . , n. A plane graph G is facial non-repetitively l-choosable if for every list assignment L : V → 2 N with minimum list size at least l there is a facial non-repetitive vertex-colouring ϕ with colours from the associated lists. The facial Thue choice number, π f l (G), of a plane graph G is the minimum number l such that G is facial non-repetitively l-choosable. We use the so-called entropy compression method to show that π f l (G) ≤ c∆ for some absolute constant c and G a plane graph with maximum degree ∆. Moreover, we give some better (constant) upper bounds on π f l (G) for special classes of plane graphs.

We obtain representations for relation algebras corresponding to certain edge colourings of complete graphs. Suitable colourings are obtained for the number of colours n up to 120, with two exceptions: n = 8 and n = 13. For n > 7 it was not known whether representations exist.