Partitioning de Bruijn Graphs into Fixed-Length Cycles for Robot Identification and Tracking

We propose a new camera-based method of robot identification, tracking and orientation estimation. The system utilises coloured lights mounted in a circle around each robot to create unique colour sequences that are observed by a camera. The number of robots that can be uniquely identified is limited by the number of colours available, $q$, the number of lights on each robot, $k$, and the number of consecutive lights the camera can see, $\ell$. For a given set of parameters, we would like to maximise the number of robots that we can use. We model this as a combinatorial problem and show that it is equivalent to finding the maximum number of disjoint $k$-cycles in the de Bruijn graph $\text{dB}(q,\ell)$. We provide several existence results that give the maximum number of cycles in $\text{dB}(q,\ell)$ in various cases. For example, we give an optimal solution when $k=q^{\ell-1}$. Another construction yields many cycles in larger de Bruijn graphs using cycles from smaller de Bruijn graphs: if $\text{dB}(q,\ell)$ can be partitioned into $k$-cycles, then $\text{dB}(q,\ell)$ can be partitioned into $tk$-cycles for any divisor $t$ of $k$. The methods used are based on finite field algebra and the combinatorics of words.Comment: 16 pages, 4 figures. Accepted for publication in Discrete Applied Mathematic

Intersection theorems on structures

All the graphs considered are simple, i.e., without loops or multiple edges. The intersection of two graphs is just the graph formed by the edges common to both of them. Let K be a family of graphs and n a positive integer. Then f(n,K) denotes the maximum number of distinct (labeled) graphs on n vertices such that the interesection of any two is in K. The authors first investigated this function in a previous paper [Combinatorics, II (Keszthely, 1976), pp. 1017–1030, North-Holland, Amsterdam, 1978; MR0519324 (80i:05062b)]. In particular, they proved that if K consists of all the subdivisions of K for given finite graphs, then f(n,K) is polynomially bounded. The present paper reviews the main results and reports on further progress, but contains no proofs. Let us mention a very recent open problem, due to the second author: Let T be the class of all graphs which contain a triangle. Prove or disprove f(n,T)=2(n2)−3. Apart from graphs, the authors consider intervals and arithmetic progressions. Those problems were introduced by R. L. Graham and the authors [J. Combin. Theory Ser. A 28 (1980), no. 1, 106–110]. Among other results, they have proved that one cannot do better than taking all the subsets of size ≤3 of the integers {1,⋯,n}, if the pairwise intersections have to be arithmetic progressions. The authors prove that if the intersection has to be an arithmetic progression of length at least k, k≥2, then the optimal bound is (π2/24+o(1))n2. The case k=1 is still open

On embeddings of CAT(0) cube complexes into products of trees

We prove that the contact graph of a 2-dimensional CAT(0) cube complex ${\bf X}$ of maximum degree $\Delta$ can be coloured with at most $\epsilon(\Delta)=M\Delta^{26}$ colours, for a fixed constant $M$. This implies that ${\bf X}$ (and the associated median graph) isometrically embeds in the Cartesian product of at most $\epsilon(\Delta)$ trees, and that the event structure whose domain is ${\bf X}$ admits a nice labeling with $\epsilon(\Delta)$ labels. On the other hand, we present an example of a 5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes which cannot be embedded into a Cartesian product of a finite number of trees. This answers in the negative a question raised independently by F. Haglund, G. Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the computation of the bounds in Theorem 1. Some figures repaire

Automatic enumeration of regular objects

We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the initial counting sequence and for asymptotic analysis. The key tool is the scalar product for symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer Sequence

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application