The (nearly) complete dog

Embedding of Complete and Nearly Complete Binary Trees into Hypercubes

A new simple algorithm for optimal embedding of complete binary trees into hypercubes as well as a node-by-node algorithm for embedding of nearly complete binary trees into hypercubes are presented.

Jacobians of nearly complete and threshold graphs

The Jacobian of a graph, also known as the Picard Group, Sandpile Group, or Critical Group, is a discrete analogue of the Jacobian of an algebraic curve. It is known that the order of the Jacobian of a graph is equal to its number of spanning trees, but the exact structure is known for only a few classes of graphs. In this paper, we compute the Jacobian for graphs of the form Kn\E(H) where H is...

Nonorientable Genus of Nearly Complete Bipartite Graphs

Saturation Numbers for Nearly Complete Graphs

An upper bound on the saturation number for graphs as well as associated extremal graphs was given by (Kászonyi and Tuza in J. Graph Theory, 10:203–210, 1986). A minor improvement of that result, which was implied in their paper, will be stated. Using this result, a series of exact saturation numbers and associated extremal graphs will be proved for the nearly complete graphs Kt − E(L), where L...

Decompositions of nearly complete digraphs into t isomorphic parts

An arc decomposition of the complete digraph DKn into t isomorphic subdigraphs is generalized to the case where the numerical divisibility condition is not satisfied. Two sets of nearly tth parts are constructively proved to be nonempty. These are the floor tth class (DKn − R)/t and the ceiling tth class (DKn + S)/t, where R and S comprise (possibly copies of) arcs whose number is the smallest ...