Generalisation : graphs and colourings

The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe

On the heterochromatic number of hypergraphs associated to geometric graphs and to matroids

The heterochromatic number hc(H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a hyperedge of H all of whose vertices have different colours. We denote by nu(H) the number of vertices of H and by tau(H) the size of the smallest set containing at least two vertices of each hyperedge of H. For a complete geometric graph G with n > 2 vertices let H = H(G) be the hypergraph whose vertices are the edges of G and whose hyperedges are the edge sets of plane spanning trees of G. We prove that if G has at most one interior vertex, then hc(H) = nu(H) - tau(H) + 2. We also show that hc(H) = nu(H) - tau(H) + 2 whenever H is a hypergraph with vertex set and hyperedge set given by the ground set and the bases of a matroid, respectively

Homogeneous colourings of graphs

summary:A proper vertex $k$-colouring of a graph $G$ is called $l$-homogeneous if the number of colours in the neigbourhood of each vertex of $G$ equals $l$. We explore basic properties (the existence and the number of used colours) of homogeneous colourings of graphs in general as well as of some specific graph families, in particular planar graphs

Coloring face hypergraphs on surfaces

AbstractThe face hypergraph of a graph G embedded on a surface has the same vertex set as G and its edges are the sets of vertices forming faces of G. A hypergraph is k-choosable if for each assignment of lists of colors of sizes k to its vertices, there is a coloring of the vertices from these lists avoiding a monochromatic edge.We prove that the face hypergraph of the triangulation of a surface of Euler genus g is O(g3)-choosable. This bound matches a previously known lower bound of order Ω (g3). If each face of the graph is incident with at least r distinct vertices, then the face hypergraph is also O(gr)-choosable. Note that colorings of face hypergraphs for r=2 correspond to usual vertex colorings and the upper bound O(g) thus follows from Heawood’s formula. Separate results for small genera are presented: the bound 3 for triangulations of the surface of Euler genus g=3 and the bound 7+36g+496 for surfaces of Euler genus g≥3. Our results dominate the previously known bounds for all genera except for g=4,7,8,9,14

Hypergraph cuts above the average

An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size m/2 + Ω(√m), and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is Ω(√m) larger than the expected size of a random r-cut. Moreover, in the case where k = 3 and r = 2 this bound is best possible and is attained by Steiner triple systems. Surprisingly, for all other cases (that is, if k ≥ 4 or r ≥ 3), we show that every m-edge k-uniform hypergraph has an r-cut whose size is Ω(m^(5/9)) larger than the expected size of a random r-cut. This is a significant difference in behaviour, since the amount by which the size of the largest cut exceeds the expected size of a random cut is now considerably larger than the standard deviation

The Complexity of Surjective Homomorphism Problems -- a Survey

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 open complexity

Homogeneous colourings of graphs

A proper vertex $k$-colouring of a graph $G$ is called $l$-homogeneous if the number of colours in the neigbourhood of each vertex of $G$ equals $l$. We explore basic properties (the existence and the number of used colours) of homogeneous colourings of graphs in general as well as of some specific graph families, in particular planar graphs