In 1975, Leech introduced the problem of finding trees whose edges can be labeled with positive integers so that the sums of the edge labels between vertices is {1, 2, · · · , (n 2 )}, where n is the number of vertices. We refer to such trees as perfect distance trees. In a previous paper, this problem was generalized to graphs in general and in particular, forests. Here, we show that tK1,2 can...

The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal o...

Perfect graphs were defined by Claude Berge in the 1960s. They are important objects for graph theory, linear programming and combinatorial optimization. Claude Berge made a conjecture about them, that was proved by Chudnovsky, Robertson, Seymour and Thomas in 2002, and is now called the strong perfect graph theorem. This is a survey about perfect graphs, mostly focused on the strong perfect gr...

The Mondrian problem consists of dissecting a square side length n∈N into non-congruent rectangles with natural sides such that the difference d(n) between largest and smallest areas partitioning is minimum. In this paper, we compute some bounds on in terms number partition. These provide us optimal partitions for values n∈N. We sequence d(n)∕n2 tends to zero n large enough. For case ‘perfect’ ...

It is shown that a class of Steiner triple systems of order 2−1, obtained by some special switchings from the Hamming Steiner triple system, is embedded into some perfect code, constructed by known switchings of ijk-components from the binary Hamming code. The number of Steiner triple systems of order n and rank less or equal n− log(n + 1) + 2, embedded into perfect binary codes of length n, is...

In perfect foresight dynamics, an action is linearly stable if expectation that people will always choose the action is self-fulfilling. A symmetric game is a PIM game if an opponent’s particular action maximizes the incentive of an action, independently of the rest of the players. This class includes supermodular games, games with linear incentives and so forth. We show that, in PIM games, lin...

The existence question for tiling of the n-dimensional Euclidian space by crosses is well known. A few existence and nonexistence results are known in the literature. Of special interest are tilings of the Euclidian space by crosses with arms of length one, also known as Lee spheres with radius one. Such a tiling forms a perfect code. In this paper crosses with arms of length half are considere...

The existence of perfect state transfer in quantum spin networks based on integral circulant graphs has been considered recently by Saxena, Severini and Shparlinski. Motivated by the mentioned work, Bašić, Petković and Stevanović give the simple condition for the characterization of integral circulant graphs allowing the perfect state transfer in terms of its eigenvalues. They stated that integ...

We say that a graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. We consider various problems concerning perfect Hpackings: Given n, r,D ∈ N, we characterise the edge density threshold that ensures a perfect Kr-packing in any graph G on n vertices and with minimum degree δ(G) ≥ D. We also give two conjectures concerning degree...