The pre-coloring extension problem consists, given a graph G and a subset of nodes to which some colors are already assigned, in nding a coloring of G with the minimum number of colors which respects the pre-coloring assignment. This can be reduced to the usual coloring problem on a certain contracted graph. We prove that pre-coloring extension is polynomial for complements of Meyniel graphs. W...

We study a graph transformation (defined by Seidel) called switching which, given a graph G = ( V,E) and a subset W C Jf of its vertices, builds a new graph by exchanging the cocycle linking W to V\ W with its complement. Switching is an equivalence relation and the associated equivalence classes are called switching classes. A switching class is perfect if it contains only perfect graphs. We s...

We propose new families of graphs which exhibit quantum perfect state transfer. Our constructions are based on the join operator on graphs, its circulant generalizations, and the Cartesian product of graphs. We build upon the results of Bašić and Petković (Applied Mathematics Letters 22(10):1609-1615, 2009) and construct new integral circulants and regular graphs with perfect state transfer. Mo...

A perfect matching of a graph is a subset of edges which covers every vertex exactly once, that is, for every vertex there is exactly one edge in the set with that vertex as endpoint. The dimer model is the study of the set of perfect matchings of a (possibly infinite) graph. The most well-known example is when the graph is Z, for which perfect matchings are equivalent (via a simple duality) to...

We obtain the characterization of saturation class and approximation spaces for perfect spline approximation. Also, a generalized perfect spline approximation is investigated.

Multicasting K independent messages via multiple-input multiple-output (MIMO) channels to multiple users where each user already has a subset of messages as side information is studied. A general framework of constructing layered space-time index coding (LSTIC) from a large class of space-time block codes (STBC), including perfect STBC, is proposed. We analyze the proposed LSTIC and show that i...

Given a set of numbers, the two-way partitioning problem is to divide them into two subsets, so that the sum of the numbers in each subset are as nearly equal as possible. The problem is NP-complete, and is contained in many scheduling applications. Based on a polynomial-time heuristic due to Karmarkar and Karp, we present a new algorithm, called Complete Karmarkar Karp (CKK), that optimally so...

A given collection of sets has a natural partial order induced by the subset relation. Let the size N of the collection be deened as the sum of the car-dinalities of the sets that comprise it. Algorithms have recently been discovered that compute the partial order in worst-case time O(N 2 = log N). This paper gives a variant implementation of a previously proposed algorithm which is shown to ha...

We discuss a proof-producing compiler for a subset of higher order logic. The translation validation is automatic, and is based on Hoare rules derived from a compositional semantics for sequences of instructions for an ARM-like machine. Partial and total correctness are dealt with. The main focus is on issues in the intermediate level and back-end of the compiler.