A set S of edge-disjoint hamilton cycles in a graph T is said to be maximal if the hamilton cycles in S form a subgraph of T such that T − E(S) has no hamilton cycle. The spectrum of a graph T is the set of integers m such that T contains a maximal set of m edge-disjoint hamilton cycles. This spectrum has previously been determined for all complete graphs, all complete bipartite graphs, and man...

Polar graphs generalise bipartite graphs, cobipartite graphs, and split graphs, and they constitute a special type of matrix partitions. A graph is polar if its vertex set can be partitioned into two, such that one part induces a complete multipartite graph and the other part induces a disjoint union of complete graphs. Deciding whether a given arbitrary graph is polar, is an NP-complete proble...

Let m ≥ 2 be an integer. We say that a poset P = (X, ) is m-partite if X has a partition X = X1 ∪ · · · ∪Xm such that (1) each Xi forms an antichain in P, and (2) x ≺ y implies x ∈ Xi and y ∈ Xj where i, j ∈ {1, . . . , m} and i < j. If P is m-partite for some m ≥ 2, then we say it is multipartite. – In this article we discuss the order dimension of multipartite posets in general and derive tig...

Polar graphs generalise bipartite, cobipartite, and split graphs, and they constitute a special type of matrix partitions. A graph is polar if its vertex set can be partitioned into two, such that one part induces a complete multipartite graph and the other part induces a disjoint union of complete graphs. Deciding whether a given arbitrary graph is polar, is an NP-complete problem. Here we sho...

A heterochromatic tree is an edge-colored tree in which any two edges have different colors. The heterochromatic tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum k such that whenever the edges of the graph G are colored with r colors, the vertices of G can be covered by at most k vertexdisjoint heterochromatic trees. In this paper we determine the heterochrom...

A new, constructive proof with a small explicit constant is given to the Erdős-Pyber theorem which says that the edges of a graph on n vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at most O(n/ logn) times. The theorem is generalized to uniform hypergraphs. Similar bounds with smaller constant value is provided for fractional partitioning both for...

We investigate continuous-variable (CV) multipartite unlockable bound entangled states. Comparing with the qubit multipartite unlockable bound entangled states, CV multipartite unlockable bound entangled states present some new and different properties. CV multipartite unlockable bound entangled states may serve as a useful quantum resource for new multiparty communication schemes. The experime...

Let G be a graph and let s be a vertex of G. We consider the structure of the set of all lifts of two edges incident with s that preserve edge-connectivity. Mader proved that two mild hypotheses imply there is at least one pair that lifts, while Frank showed (with the same hypotheses) that there are at least (deg(s) − 1)/2 disjoint pairs that lift. We consider the lifting graph: its vertices ar...