ARTICLES

**Title:**(k,g)-cages are 3-connected

**Authors:**Michael Daven

**Periodical:***Discrete Mathematics*, v199, 1999, p207-215

**Title:**The Johnson graph has connectivity $\delta$

**Authors:**Michael Daven

**Periodical:***Proceedings of the 30th Southeastern International Conference on Combinatorics, Graph Theory, & Computing*, Boca Raton, 1999, p123-128

**Title:**Maximal sets of hamilton cycles in complete multipartite graphs

**Authors:**Michael Daven

**Periodical:***Discrete Mathematics*, v43, 2003, p49-66

**Title:**Multidesigns for graph-pairs of order 4 and 5

**Authors:**Michael Daven

**Periodical:***Graphs & Combinatorics*, v19, n4, 2003, p433-447

**Absgtract:**The graph decomposition problem is well known. We say a subgraph G divides K m if the edges of K m can be partitioned into copies of G. Such a partition is called a G-decomposition or G-design. The graph multidecomposition problem is a variation of the above. By a graph-pair of order t, we mean two non-isomorphic graphs G and H on t non-isolated vertices for which GHK t for some integer t4. Given a graph-pair (G,H), if the edges of K m can be partitioned into copies of G and H with at least one copy of G and one copy of H, we say (G,H) divides K m . We will refer to this partition as a (G,H)-multidecomposition. In this paper, we consider the existence of multidecompositions for several graph-pairs. For the pairs (G,H) which satisfy GHK 4 or K 5, we completely determine the values of m for which K m admits a (G,H)-multidecomposition. When K m does not admit a (G,H)-multidecomposition, we instead find a maximum multipacking and a minimum multicovering. A multidesign is a multidecomposition, a maximum multipacking, or a minimum multicovering.

**Title:**Multidecompositions of the complete graph

**Authors:**Michael Daven

**Periodical:***Ars Combinitoria*, vLXXII, 2004, p17-22

**Title:**Multidesigns of the $\lambda$-fold complete graph for graph-pairs of orders 4 and 5

**Authors:**Michael Daven

**Periodical:***Australasian Journal of Combinatorics*,
v32, June 2005, p125-136

**Title:**Multidesigns for graph-triples of order 6

**Authors:**Michael Daven

**Periodical:***Congressus Numerantium*,
v183, 2006, p139-160

**Title:**Math trails in undergraduate mathematics

**Authors:**Michael Daven, Lee Fothergill

**Periodical:***New York State Mathematics Teachers Journal*,
v61, 2011, p32-38

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