4,992 research outputs found
Packings and Coverings of Complete Graphs with a Hole with the 4-Cycle with a Pendant Edge
In this thesis, we consider packings and coverings of various complete graphs with the 4-cycle with a pendant edge. We consider both restricted and unrestricted coverings. Necessary and sufficient conditions are given for such structures for (1) complete graphs Kv, (2) complete bipartite graphs Km,n, and (3) complete graphs with a hole K(v,w)
Tremain equiangular tight frames
Equiangular tight frames provide optimal packings of lines through the
origin. We combine Steiner triple systems with Hadamard matrices to produce a
new infinite family of equiangular tight frames. This in turn leads to new
constructions of strongly regular graphs and distance-regular antipodal covers
of the complete graph.Comment: 11 page
Tremain Equiangular Tight Frames
Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph
Perfect packings with complete graphs minus an edge
Let K_r^- denote the graph obtained from K_r by deleting one edge. We show
that for every integer r\ge 4 there exists an integer n_0=n_0(r) such that
every graph G whose order n\ge n_0 is divisible by r and whose minimum degree
is at least (1-1/chi_{cr}(K_r^-))n contains a perfect K_r^- packing, i.e. a
collection of disjoint copies of K_r^- which covers all vertices of G. Here
chi_{cr}(K_r^-)=r(r-2)/(r-1) is the critical chromatic number of K_r^-. The
bound on the minimum degree is best possible and confirms a conjecture of
Kawarabayashi for large n
An Ore-type theorem for perfect packings in graphs
We say that a graph G has a perfect H-packing (also called an H-factor) if
there exists a set of disjoint copies of H in G which together cover all the
vertices of G. Given a graph H, we determine, asymptotically, the Ore-type
degree condition which ensures that a graph G has a perfect H-packing. More
precisely, let \delta_{\rm Ore} (H,n) be the smallest number k such that every
graph G whose order n is divisible by |H| and with d(x)+d(y)\geq k for all
non-adjacent x \not = y \in V(G) contains a perfect H-packing. We determine
\lim_{n\to \infty} \delta_{\rm Ore} (H,n)/n.Comment: 23 pages, 1 figure. Extra examples and a sketch proof of Theorem 4
added. To appear in the SIAM Journal on Discrete Mathematic
Mathematical optimization for packing problems
During the last few years several new results on packing problems were
obtained using a blend of tools from semidefinite optimization, polynomial
optimization, and harmonic analysis. We survey some of these results and the
techniques involved, concentrating on geometric packing problems such as the
sphere-packing problem or the problem of packing regular tetrahedra in R^3.Comment: 17 pages, written for the SIAG/OPT Views-and-News, (v2) some updates
and correction
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