André Kündgen

Coloring Face-Hypergraphs of Graphs on Surfaces

The face-hypergraph, H(G), of a graph G embedded in a surface has vertex set V(G), and every face of G corresponds to an edge of H(G) consisting of the vertices incident to the face. We study coloring parameters of these embedded hypergraphs. A hypergraph is k-colorable (k-choosable) if there is a coloring of its vertices from a set of k colors (from every assignment of lists of size k to its vertices) such that no edge is monochromatic. Thus a proper coloring of a face-hypergraph corresponds to a vertex coloring of the underlying graph such that no face is monochromatic. We show that hypergraphs can be extended to face-hypergraphs in a natural way and use tools from topological graph theory, the theory of hypergraphs, and design theory to obtain general bounds for the coloring and choosability problems. To show the sharpness of several bounds, we construct for every even n an n-vertex pseudo-triangulation of a surface such that every triple is a face exactly once. We provide supporting evidence for our conjecture that every plane face-hypergraph is 2-choosable and we pose several open questions, most notably: Can the vertices of a planar graph always be properly colored from lists of size 4, with the restriction on the lists that the colors come in pairs and a color is in a list if and only if its twin color is? An affirmative answer to this question would imply our conjecture, as well as the Four Color Theorem and several open problems.

On a Generalized Anti-Ramsey Problem

For positive integers , a coloring of is called a -coloring if the edges of every receive at least and at most colors. Let denote the maximum number of colors in a -coloring of . Given we determine the largest such that all -colorings of have at most O(n) colors and we determine asymptotically when it is of order equal to . We give several bounds and constructions.

Covering cliques with spanning bicliques

We give counterexamples to two conjectures of Bill Jackson in Some remarks on arc-connectivity, vertex splitting, and orientation in graphs and digraphs (Journal of Graph Theory 12(3):429–436, 1988) concerning orientations of mixed graphs and splitting off in digraphs, and prove the first conjecture in the (di-) Eulerian case(s). Beside that we solve a degree constrained non-uniform directed augmentation problem for di-Eulerian mixed graphs.

On decompositions of complete hypergraphs

We study the minimum number of complete r-partite r-uniform hypergraphs needed to partition the edges of the complete r-uniform hypergraph on n vertices and we improve previous results of Alon.

Bipartite anti-Ramsey numbers of cycles: BIPARTITE ANTI-RAMSEY NUMBERS

We determine the maximum number of colors in a coloring of the edges of Km;n such that every cycle of length 2k contains at least two edges of the same color. One of our main tools is a result on generalized path covers in balanced bipartite graphs. For positive integers q a, let g(a;q) be the maximum number of edges in a spanning subgraph G of Ka;a such that the minimum number of vertex-disjoint even paths and pairs of vertices from distinct partite sets needed to cover V (G) is q. We prove that g(a,q)1⁄4 a aq þmax {a; 2q 2} . 2004 Wiley Periodicals, Inc. J Graph Theory 47: 9–28,

Turán problems for integer-weighted graphs: TURÁN PROBLEMS FOR INTEGER-WEIGHTED GRAPHS

A multigraph is (k, r)-dense if every k-set spans at most r edges. What is the maximum number of edges exN(n, k, r) in a (k, r)-dense multigraph on n vertices? We determine the maximum possible weight of such graphs for almost all k and r (e.g., for all r > k) by determining a constant m = m(k, r) and showing that exN(n, k, r) = m ( n 2 ) + O(n), thus giving a generalization of Turán’s theorem. We find exact answers in many cases, even when negative integer weights are also allowed. In fact, our main result is to determine the maximum weight of (k, r)-dense n-vertex multigraphs with arbitrary integer weights with an O(n) error term. c

Covering a graph with cuts of minimum total size

Abstract A cut in a graph G is the set of all edges between some set of vertices S and its complement S =V(G)−S . A cut-cover of G is a collection of cuts whose union is E ( G ) and the total size of a cut-cover is the sum of the number of edges of the cuts in the cover. The cut-cover size of a graph G, denoted by cs( G ), is the minimum total size of a cut-cover of G. We give general bounds on cs( G ), find sharp bounds for classes of graphs such as 4-colorable graphs and random graphs. We also address algorithmic aspects and give sharp bounds for the sum of the cut-cover sizes of a graph and its complement. We close with a list of open problems.

Minimal Completely Separating Systems of k-Sets

Let n and k be fixed positive integers. A collection C of k-sets of [n] is a completely separating system if, for all distinct i, j?[n], there is an S?C for which i?S and j?S. Let R(n, k) denote the minimum size of such a C. Our results include showing that if nk is a sequence with k?nk?k1+? for every ?>0, then[formula]

On 2-factors with long cycles in cubic graphs

Every 2-connected cubic graph G has a 2-factor, and much effort has gone into studying conditions that guarantee G to be Hamiltonian. We show that if G is not Hamiltonian, then G is either the Petersen graph or contains a 2-factor with a cycle of length at least 7. We also give infinite families of, respectively, 2- and 3-connected cubic graphs in which every 2-factor consists of cycles of length at most, respectively, 10 and 16.

Switchings, extensions, and reductions in central digraphs

A directed graph is called central if its adjacency matrix A satisfies the equation A^2=J, where J is the matrix with a 1 in each entry. It has been conjectured that every central directed graph can be obtained from a standard example by a sequence of simple operations called switchings, and also that it can be obtained from a smaller one by an extension. We disprove these conjectures and present a general extension result which, in particular, shows that each counterexample extends to an infinite family.