Diameter Perfect Lee Codes

Lee codes have been intensively studied for more than 40 years. Interest in these codes has been triggered by the Golomb-Welch conjecture on the existence of the perfect error-correcting Lee codes. In this paper we deal with the existence and enumeration of diameter perfect Lee codes. As main results we determine all $q$ for which there exists a linear diameter-4 perfect Lee code of word length $n$ over $Z_{q},$ and prove that for each $n\geq 3$ there are uncountable many diameter-4 perfect Lee codes of word length $n$ over $Z.$ This is in a strict contrast with perfect error-correcting Lee codes of word length $n$ over $Z\,$\ as there is a unique such code for $n=3,$ and its is conjectured that this is always the case when $2n+1$ is a prime. We produce diameter perfect Lee codes by an algebraic construction that is based on a group homomorphism. This will allow us to design an efficient algorithm for their decoding. We hope that this construction will turn out to be useful far beyond the scope of this paper

Fullerene graphs of small diameter

A fullerene graph is a cubic bridgeless plane graph with only pentagonal and hexagonal faces. We exhibit an infinite family of fullerene graphs of diameter $\sqrt{4n/3}$, where $n$ is the number of vertices. This disproves a conjecture of Andova and \v{S}krekovski [MATCH Commun. Math. Comput. Chem. 70 (2013) 205-220], who conjectured that every fullerene graph on $n$ vertices has diameter at least $\lfloor \sqrt{5n/3}\rfloor-1$

Diameter two properties, convexity and smoothness

We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties. We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizations of the following property for a Banach space $X$: "For every slice $S$ of $B_X$ and every norm-one element $x$ in $S$, there is a point $y\in S$ in distance as close to 2 as we want." Spaces with this property are shown to have non-smooth bidual.Comment: Removed Proposition 2.7 from version [v1] because of a gap in the proof. arXiv admin note: text overlap with arXiv:1506.0523

Augmenting graphs to minimize the diameter

We study the problem of augmenting a weighted graph by inserting edges of bounded total cost while minimizing the diameter of the augmented graph. Our main result is an FPT 4-approximation algorithm for the problem.Comment: 15 pages, 3 figure

The diameter of weighted random graphs

In this paper we study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight paths between all vertices (the weighted diameter) of sparse random graphs, when the edge weights are i.i.d. exponential random variables.Comment: Published at http://dx.doi.org/10.1214/14-AAP1034 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org