Mixing graph colourings

This thesis investigates some problems related to graph colouring, or, more precisely, graph re-colouring. Informally, the basic question addressed can be phrased as follows. Suppose one is given a graph G whose vertices can be properly k-coloured, for some k ≥ 2. Is it possible to transform any k-colouring of G into any other by recolouring vertices of G one at a time, making sure a proper k-colouring of G is always maintained? If the answer is in the affirmative, G is said to be k-mixing. The related problem of deciding whether, given two k-colourings of G, it is possible to transform one into the other by recolouring vertices one at a time, always maintaining a proper k-colouring of G, is also considered. These questions can be considered as having a bearing on certain mathematical and ‘real-world’ problems. In particular, being able to recolour any colouring of a given graph to any other colouring is a necessary pre-requisite for the method of sampling colourings known as Glauber dynamics. The results presented in this thesis may also find application in the context of frequency reassignment: given that the problem of assigning radio frequencies in a wireless communications network is often modelled as a graph colouring problem, the task of re-assigning frequencies in such a network can be thought of as a graph recolouring problem. Throughout the thesis, the emphasis is on the algorithmic aspects and the computational complexity of the questions described above. In other words, how easily, in terms of computational resources used, can they be answered? Strong results are obtained for the k = 3 case of the first question, where a characterisation theorem for 3-mixing graphs is given. For the second question, a dichotomy theorem for the complexity of the problem is proved: the problem is solvable in polynomial time for k ≤ 3 and PSPACE-complete for k ≥ 4. In addition, the possible length of a shortest sequence of recolourings between two colourings is investigated, and an interesting connection between the tractability of the problem and its underlying structure is established. Some variants of the above problems are also explored

Connectedness of the graph of vertex-colourings

For a positive integer k and a graph G, the k-colour graph of G , Ck(G), is the graph that has the proper k-vertex-colourings of G as its vertex set, and two k -colourings are joined by an edge in Ck(G) if they differ in colour on just one vertex of G . In this note some results on the connectedness of Ck(G) are proved. In particular, it is shown that if G has chromatic number k∈{2,3}, then Ck(G) is not connected. On the other hand, for k⩾4 there are graphs with chromatic number k for which Ck(G) is not connected, and there are k -chromatic graphs for which Ck(G) is connected

Finding Paths between Graph Colourings: PSPACE-completeness and Superpolynomial Distances

Suppose we are given a graph G together with two proper vertex k-colourings of G, α and β. How easily can we decide whether it is possible to transform α into β by recolouring vertices of G one at a time, making sure we always have a proper k-colouring of G? This decision problem is trivial for k=2, and decidable in polynomial time for k=3. Here we prove it is PSPACE-complete for all k >= 4. In particular, we prove that the problem remains PSPACE-complete for bipartite graphs, as well as for: (i) planar graphs and 4 = 4, a class of graphs {GN,k:N>0}, together with two k-colourings for each GN,k, such that the minimum number of recolouring steps required to transform the first colouring into the second is superpolynomial in the size of the graph: the minimum number of steps is Ω(2N), whereas the size of GN is O(N2). This is in stark contrast to the k=3 case, where it is known that the minimum number of recolouring steps is at most quadratic in the number of vertices. We also show that a class of bipartite graphs can be constructed with this property, and that: (i) for 4 <= k <= 6 planar graphs and (ii) for k=4 bipartite planar graphs can be constructed with this property. This provides a remarkable correspondence between the tractability of the problem and its underlying structure

Violencia en la niñez y autoestima en estudiantes de un colegio alternativo de la ciudad de Arequipa, 2022

La investigación tuvo como objetivo general determinar la relación entre la violencia en la niñez y la autoestima en estudiantes de un colegio alternativo de la ciudad de Arequipa, 2022. La metodología correspondió a un enfoque cuantitativo, de tipo básica y alcance correlacional, se conformó con una muestra de 47 estudiantes a quienes les fue aplicado un cuestionario como instrumento de recolección. El principal resultado encontrado evidenció un Rho de Spearman = -0.583, significando que existe una correlación negativa considerable entre la violencia en la niñez y la autoestima en estudiantes de un colegio alternativo de la ciudad de Arequipa. Además, se halló una significancia bilateral de 0,000, que es <0.05, por tanto, se acepta la Ha y rechaza la Ho; es decir, existe relación inversa significativa entre la violencia en la niñez y la autoestima en estudiantes de un colegio alternativo de la ciudad de Arequipa, 2022. En consecuencia, se concluyó que cuanto mayor sea la violencia en la niñez, menor será la autoestima en los estudiantes del colegio alternativo, ya que violencia en la niñez afecta de forma considerable en la autoestima de los estudiantes

Averaging sums of powers of integers and Faulhaber polynomials

As an application of Faulhaber’s theorem on sums of powers of integers and the associated Faulhaber polynomials, in this article we provide the solution to the following two questions: (1) when is the average of sums of powers of integers itself a sum of the first n integers raised to a power? and (2), when is the average of sums of powers of integers itself a sum of the first n integers raised to a power, times the sum of the first n squares? In addition to this, we derive a family of recursion formulae for the Bernoulli numbers. Keywords: sums of powers of integers, Faulhaber polynomials, matrix inversion, Bernoulli number