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A decomposition of λ copies of monochromatic K v into copies of K 4 such that each copy of K 4 contains at most one edge from each K v is called a proper edge coloring of a BIBD(v, 4, λ). We show that the necessary conditions are sufficient for the existence of a BIBD(v, 4, λ) which has such a proper edge coloring.
The Hamilton-Waterloo Problem (HWP) in the case of Cm-factors and Cn-factors asks whether Kv, where v is odd (or Kv − F , where F is a 1-factor and v is even), can be decomposed into r copies of a 2-factor made either entirely of m-cycles and s copies of a 2-factor made entirely of n-cycles. In this paper, we give some general constructions for such ∗ This… (More)
Date Acknowledgements I would like to dedicate this first to Nancy Lachapelle for her everlasting support and love. Thank you darling. Thank you Mom and Dad for all the visits and trips to Walmart and the support you have given me. They did not go unnoticed. Dave Kamin, I wish you could be here to right now. It has been too long since our last talk. I am… (More)
In this paper we introduce and study a new graph invariant derived from the degree sequence of a graph G, called the sub-k-domination number and denoted sub k (G). We show that sub k (G) is a computationally efficient sharp lower bound on the k-domination number of G, and improves on several known lower bounds. We also characterize the sub-k-domination… (More)