arXiv subjects

Robert F. Bailey

Publications and source records attributed to Robert F. Bailey.

22 records · Page 2Linked to original sources

arXiv2011

On the metric dimension of Grassmann graphs

The {\em metric dimension} of a graph $Γ$ is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. We consider the Grassmann graph $G_q(n,k)$ (whose vertices are the $k$-subspaces of $\mathbb{F}_q^n$, and are adjacent if they intersect in a $(k-1)$-subspace) for $k\geq 2$, and find a constructive upper bound on its metric dimension. Our bound is equal to the number of 1-dimensional subspaces of $\mathbb{F}_q^n$.

math.CO↗

arXiv2011

Generalized packing designs

Generalized $t$-designs, which form a common generalization of objects such as $t$-designs, resolvable designs and orthogonal arrays, were defined by Cameron [P.J. Cameron, A generalisation of $t$-designs, \emph{Discrete Math.}\ {\bf 309} (2009), 4835--4842]. In this paper, we define a related class of combinatorial designs which simultaneously generalize packing designs and packing arrays. We describe the sometimes surprising connections which these generalized designs have with various known classes of combinatorial designs, including Howell designs, partial Latin squares and several classes of triple systems, and also concepts such as resolvability and block colouring of ordinary designs and packings, and orthogonal resolutions and colourings. Moreover, we derive bounds on the size of a generalized packing design and construct optimal generalized packings in certain cases. In particular, we provide methods for constructing maximum generalized packings with $t=2$ and block size $k=3$ or 4.

math.CO↗

arXiv2011

Uncoverings on graphs and network reliability

We propose a network protocol similar to the $k$-tree protocol of Itai and Rodeh [{\em Inform.\ and Comput.}\ {\bf 79} (1988), 43--59]. To do this, we define an {\em $t$-uncovering-by-bases} for a connected graph $G$ to be a collection $\mathcal{U}$ of spanning trees for $G$ such that any $t$-subset of edges of $G$ is disjoint from at least one tree in $\mathcal{U}$, where $t$ is some integer strictly less than the edge connectivity of $G$. We construct examples of these for some infinite families of graphs. Many of these infinite families utilise factorisations or decompositions of graphs. In every case the size of the uncovering-by-bases is no larger than the number of edges in the graph and we conjecture that this may be true in general.

math.CO↗

arXiv2011

Generalized covering designs and clique coverings

Inspired by the "generalized t-designs" defined by Cameron [P. J. Cameron, A generalisation of t-designs, Discrete Math. 309 (2009), 4835--4842], we define a new class of combinatorial designs which simultaneously provide a generalization of both covering designs and covering arrays. We then obtain a number of bounds on the minimum sizes of these designs, and describe some methods of constructing them, which in some cases we prove are optimal. Many of our results are obtained from an interpretation of these designs in terms of clique coverings of graphs.

math.CO↗