Martin Loebl

The chromatic polynomial of fatgraphs and its categorification

Abstract Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homology to show that the torsion-free part of our chromatic homology is independent of the choice of planar embedding of a graph. We extend our construction and categorify the Bollobas–Riordan polynomial (a generalization of the Tutte polynomial to embedded graphs). We prove that both our chromatic homology and the Khovanov homology of an associated link can be recovered from this categorification.

Efficient Subgraph Packing

We are concerned with a natural generalization of the matching problem: the packing of graphs from a pregiven family. We characterize completely the complexity of deciding the existence of a perfect packing, if the pregiven family consists of two graphs, one of which is one edge on two vertices. Our results complete the work of Cornuejols. Hartvigsen, and Pulleyblank, and Hell and Kirkpatrick.

On matroids induced by packing subgraphs

Abstract The main result of the paper is a characterization of connected graphs H with the property: For any graph G, the subsets of vertices that can be saturated by packing edges of G and copies of H are independent sets of a matroid.

Optimization via enumeration: a new algorithm for the Max Cut Problem

Abstract.We present a polynomial time algorithm to find the maximum weight of an edge-cut in graphs embeddable on an arbitrary orientable surface, with integral weights bounded in the absolute value by a polynomial of the size of the graph.</The algorithm has been implemented for toroidal grids using modular arithmetics and the generalized nested dissection method. The applications in statistical physics are discussed.

Subgraph Packing — A Survey

The research on factors of graphs concentrated mostly on factors satisfying certain local degree conditions, like regular factors or factors with degrees within prescribed intervals. More recently, also other kinds of factors have been investigated. Here we survey results on factors with prescribed components. Let F be a family of graphs. A graph G is said to have an F — factor if it has a factor each component of which is isomorphic to a member of family F The F — factor problem is to decide whether a given graph G admits an F — factor. The F — packing problem is to find a subgraph of maximum order which admits an F — factor.

Jordan graphs

erties which make their use advantageous whenever the underlying situation allows us to use them.  1996 Academic Press, Inc. Early development of digital topology concentrated on tessellations of the plane into square-shaped pixels, each one of which had a 1 or a 0 assigned to it. It became quickly apparent that 1. MOTIVATION AND BACKGROUND in order to avoid some ‘‘paradoxes’’ one needs to consider adjacencies in addition to that provided by the edge-adjacency. Jordan curves have an interior and an exterior which The customary choice became the use of a pair of adjacencies; one for the 1-pixels and another for the 0-pixels. While this between them contain all the points not on the curve and approach can be treated in a mathematically rigorous fashion, both of which are connected (for any two interior points it nevertheless remains attractive to consider those digital there is a path from one to the other which lies entirely spaces in which a single adjacency sufficies to usefully define in the interior and for any two exterior points there is a connectivity. This not only makes the resulting mathematics path from one to the other which lies entirely in the extemore elegant, but it also brings the subject nearer to classical rior), but are disconnected from each other (every path graph theory and its enormous wealth of results. This is what from an interior point to an exterior point must contain a motivated us to search for an appropriate new concept. Jordan point on the curve). Generalization to surfaces in digital curves have an interior and an exterior which between them spaces is useful when displaying (only the exterior of a contain all the points not on the curve and both of which are surface is visible from any direction) or analyzing (the connected, but are disconnected from each other. Generalizainterior volume is well defined). An example of a Jordan tion to surfaces in digital spaces is useful when displaying (only the exterior of a surface is visible from any direction) or analyzsurface in the medical imaging application area is shown ing (the interior volume is well defined). Boundaries in digital in Fig. 1. Boundaries in digital spaces may or may not be spaces may or may not be Jordan, but are ‘‘guaranteed’’ to be Jordan (we will show examples), but are ‘‘guaranteed’’ to Jordan in spaces we call strong Jordan graphs. In this paper be Jordan in those digital spaces which we will call (strong) we define such a notion of a strong Jordan graph. We show Jordan graphs. that some previously studied classes (such as those of 1-simply We now give the necessary background definitions and connected digital spaces and of bridged graphs) are subclasses results from [1–3]. For some of the terms used, the primary of strong Jordan graphs. We also define Jordan graphs, as definitions given in these papers are not identical. Howthose digital spaces in which finite boundaries are guaranteed ever, it is proved in [3] that the defined concepts themselves to be Jordan and show that there are Jordan graphs which are are the same and so it is valid to quote any of the results not strong Jordan graphs. (Strong) Jordan graphs are characfrom these papers. terized by the existence of ‘‘cuts’’ in associated graphs, by the acyclic nature of the adjacency graphs of binary pictures, and A digital space is a pair (V, f), where V is a nonempty also by the connectedness of the immediate interiors (or exteriset (of spels) and f is a symmetric binary relation (called ors) of certain types of surfaces. The surfaces of the last category the proto-adjacency) on V, under which V is connected; include the so-called minimally near-Jordan surfaces, which that is, f is a set of ordered pairs of elements of V such that are shown to be of some interest. Finally, we tie some of these notions to standard notions of graph theory, such as the notion (i) if (c, d) [ f, then (d, c) [ f and of a separating set. Our overall conclusion is that those digital (ii) for any c and d in V, there exists a finite sequence spaces which are Jordan graphs have some very desirable propc(0), . . . , c(m) of elements of V such that m

An unprovable Ramsey-type theorem

0, c(0) 5 c, c(m) 5 d, and (c(k), c(k11)) [ f, for 0 # k , m.

Towards a theory of frustrated degeneracy

We present a new proof of the Paris-Harrington unprovable (in PA) version of Ramseys theorem. This also yields a particularly short proof of the Ketonen-Solovay result on rapidly growing Ramsey functions

Epidemiography II. Games with a dozing yet winning player

We study the question of how degeneracy of finite toroidal and 3-dimensional spin glasses depends on frustration. Using the transfer matrix method we obtain lower and upper bounds for maximum degeneracy in both 2-dimensional and 3-dimensional lattices.

Discrete Mathematics in Statistical Physics

Abstract Epidemiography designates a class of games played on directed graphs. At step k of the game, the move made on a digraph G is replicated onto f ( k ) isomorphic copies of G . The player first unable to move is the loser; his opponent the winner. We show that if G is finite and acyclic, then the game terminates for every function f : Z + → Z 0 , and we construct classes of digraphs and functions f for which the first (second) player can win. Many epidemiography games are robust : the outcome depends only on the “foliage” of G , and the winner can play randomly during much of the time. Play is very long even if f grows only linearly. Bounds on the length of play are provided.