Enclosings of Decompositions of Complete Multigraphs in 22-Edge-Connected rr-Factorizations

A decomposition of a multigraph $G$ is a partition of its edges into subgraphs $G(1), \ldots , G(k)$. It is called an $r$-factorization if every $G(i)$ is $r$-regular and spanning. If $G$ is a subgraph of $H$, a decomposition of $G$ is said to be enclosed in a decomposition of $H$ if, for every $1 \leq i \leq k$, $G(i)$ is a subgraph of $H(i)$. Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of $\lambda K_n$ to be enclosed in some $2$-edge-connected $r$-factorization of $\mu K_{m}$ for some range of values for the parameters $n$, $m$, $\lambda$, $\mu$, $r$: $r=2$, $\mu>\lambda$ and either $m \geq 2n-1$, or $m=2n-2$ and $\mu = 2$ and $\lambda=1$, or $n=3$ and $m=4$. We generalize their result to every $r \geq 2$ and $m \geq 2n - 2$. We also give some sufficient conditions for enclosing a given decomposition of $\lambda K_n$ in some $2$-edge-connected $r$-factorization of $\mu K_{m}$ for every $r \geq 3$ and $m = (2 - C)n$, where $C$ is a constant that depends only on $r$, $\lambda$ and~$\mu$.Comment: 17 pages; fixed the proof of Theorem 1.4 and other minor change

Convex Cone Conditions on the Structure of Designs

Various known and original inequalities concerning the structure of combinatorial designs are established using polyhedral cones generated by incidence matrices. This work begins by giving definitions and elementary facts concerning t-designs. A connection with the incidence matrix W of t-subsets versus k-subsets of a finite set is mentioned. The opening chapter also discusses relevant facts about convex geometry (in particular, the Farkas Lemma) and presents an arsenal of binomial identities. The purpose of Chapter 2 is to study the cone generated by columns of W, viewed as an increasing union of cones with certain invariant automorphisms. The two subsequent chapters derive inequalities on block density and intersection patterns in t-designs. Chapter 5 outlines generalizations of W which correspond to hypergraph designs and poset designs. To conclude, an easy consequence of this theory for orthogonal arrays is used in a computing application which generalizes the method of two-point based samplin

Enclosings of λ-fold 5-cycle systems for u = 2

© 2014 Elsevier B.V.All rights reserved. A k-cycle system of a multigraph G is an ordered pair (V,C) where V is the vertex set of G and C is a set of k-cycles, the edges of which partition the edges of G. A k-cycle system of λKvis known as a λ-fold k-cycle system of order v. A k-cycle system of λKv(V,C) is said to be enclosed in a k-cycle system of (λ+m)Kv+v+u(V∪U,P) if C⋯P and u,m≥1. In this paper the enclosing problem for 5-cycle systems is settled in the general situation where the three parameters λ, m, and v are allowed to vary freely and u is constrained to the difficult case of adding two vertices. New graph theoretic approaches are introduced to handle this situation developing an avenue of research that is of interest in its own right