Colouring powers of cycles from random lists

Let $C_n^k$ be the $k$-th power of a cycle on $n$ vertices (i.e. the vertices of $C_n^k$ are those of the $n$-cycle, and two vertices are connected by an edge if their distance along the cycle is at most $k$). For each vertex draw uniformly at random a subset of size $c$ from a base set $S$ of size $s=s(n)$. In this paper we solve the problem of determining the asymptotic probability of the existence of a proper colouring from the lists for all fixed values of $c,k$, and growing $n$.Comment: 7 page

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

Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)

We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography, comments are welcome, published as a Dynamic Survey in Electronic Journal of Combinatoric

Two Combinatorial Models with identical Statics yet different Dynamics

Motivated by the problem of sorting, we introduce two simple combinatorial models with distinct Hamiltonians yet identical spectra (and hence partition function) and show that the local dynamics of these models are very different. After a deep quench, one model slowly relaxes to the sorted state whereas the other model becomes blocked by the presence of stable local minima.Comment: 23 pages, 11 figure

A general approach to transversal versions of Dirac-type theorems

Given a collection of hypergraphs (Formula presented.) with the same vertex set, an (Formula presented.) -edge graph (Formula presented.) is a transversal if there is a bijection (Formula presented.) such that (Formula presented.) for each (Formula presented.). How large does the minimum degree of each (Formula presented.) need to be so that (Formula presented.) necessarily contains a copy of (Formula presented.) that is a transversal? Each (Formula presented.) in the collection could be the same hypergraph, hence the minimum degree of each (Formula presented.) needs to be large enough to ensure that (Formula presented.). Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020) 498–504), a growing body of work has shown that in many cases this lower bound is tight. In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac-type results for (powers of) Hamilton cycles. For example, we derive that any collection of (Formula presented.) graphs on an (Formula presented.) -vertex set, each with minimum degree at least (Formula presented.), contains a transversal copy of the (Formula presented.) th power of a Hamilton cycle. This can be viewed as a rainbow version of the Pósa–Seymour conjecture