arXiv subjects
Publications and source records attributed to Michal Adamaszek.
The join operad arises from the combinatorial study of the iterated join of simplices. We study a suitable simplicial version of this operad which includes the symmetries given by permutations of the factors of the join. From this combinatorics we construct an E-infinity operad which coacts naturally on the chains of a simplicial set.
We investigate families of two-dimensional simplicial complexes defined in terms of vertex decompositions. They include nonevasive complexes, strongly collapsible complexes of Barmak and Miniam and analogues of 2-trees of Harary and Palmer. We investigate the complexity of recognition problems for those families and some of their combinatorial properties. Certain results follow from analogous decomposition techniques for graphs. For example, we prove that it is NP-complete to decide if a graph can be reduced to a discrete graph by a sequence of removals of vertices of degree 3.
We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known that the space of real algebraic maps is a dense subset of the space of all continuous maps. Our first result shows that, for this class of varieties, the inclusion is also a homotopy equivalence. After proving this, we restrict the class of varieties to real projective spaces. In this case, the space of algebraic maps has a ` minimum degree\rq filtration by finite dimensional subspaces and it is natural to expect that the homotopy types of the terms of the filtration approximate closer and closer the homotopy type of the space of continuous mappings as the degree increases. We prove this and compute the lower bounds of this approximation for ` even\rq components of these spaces (more precisely, we prove a very similar and closely related result, and state this one as a conjecture).
We prove that if two graphs of girth at least 6 have isomorphic squares, then the graphs themselves are isomorphic. This is the best possible extension of the results of Ross and Harary on trees and the results of Farzad et al. on graphs of girth at least 7. We also make a remark on reconstruction of graphs from their higher powers.
We study the problem of recognizing graph powers and computing roots of graphs. We provide a polynomial time recognition algorithm for r-th powers of graphs of girth at least 2r+3, thus improving a bound conjectured by Farzad et al. (STACS 2009). Our algorithm also finds all r-th roots of a given graph that have girth at least 2r+3 and no degree one vertices, which is a step towards a recent conjecture of Levenshtein that such root should be unique. On the negative side, we prove that recognition becomes an NP-complete problem when the bound on girth is about twice smaller. Similar results have so far only been attempted for r=2,3.
We investigate the structure of the currencies (systems of coins) for which the greedy change-making algorithm always finds an optimal solution (that is, a one with minimum number of coins). We present a series of necessary conditions that must be satisfied by the values of coins in such systems. We also uncover some relations between such currencies and their sub-currencies.
A graceful n-permutation is a graceful labeling of an n-vertex path P_n. In this paper we improve the asymptotic lower bound on the number of such permutations from (5/3)^n to 2.37^n. This is a computer-assisted proof based on an effective algorithm that enumerates graceful n-permutations. Our algorithm is also presented in detail.