Prove that in fact ch() = 3. Hint: Use a case distinction according to whether or not two color lists for vertices in the same part of the bipartition have a color in common. By a result of [ErRT80], there are bipartite graphs with an arbitrarily large choosability; thus k-choosability can indeed be a much […]

Let G be the digraph given in Figure 10.2 with capacity constraints b and c. We require a feasible circulation on G. By Theorem 10.2.1, we have to determine a maximal flow for the network N shown in Figure 10.3. In general, we would use one of the algorithms of Chapter 6 for such a

Let G be a mixed multi graph. Find necessary and sufficient conditions for the existence of an Euler tour in G; cf. Exercise 10.1.6. Let N = (G, b, c, s, t) be a flow network with a nonnegative lower capacity b. Describe a technique for determining a minimal feasible flow on N (that is,

Let G be a connected digraph with capacity constraints b and c, where b(e) is always positive and c(e) = ∞ for all edges e. Show that G has a feasible circulation if and only if it is strongly connected. Moreover, give a criterion for the existence of a feasible flow if we also specify

Let G be a 3-regular graph without bridges. Show that G has a perfect matching [Pet91]. Does this also hold for 3-regular graphs containing bridges? Does a 3-regular graph without bridges necessarily have a 1-factorization? Let G = (V,E) be a digraph having n vertices, m edges, and p connected components. Let M be the

Let us show that the labelling algorithm of Ford and Fulkerson (Algorithm 6.1.7) may be viewed as a special case of Algorithm 10.3.13. We choose for e0 the return arc r = ts introduced in Example 10.1.1, and color e0 black. The remaining edges e are colored as follows: black if e is void, green if e is saturated, and red otherwise. It should be clear that

Let us show that the labelling algorithm of Ford and Fulkerson (Algorithm 6.1.7) may be viewed as a special case of Algorithm 10.3.13. We choose for e0 the return arc r = ts introduced in Example 10.1.1, and color e0 black. The remaining edges e are colored as follows: black if e is void, green if e is saturated, and red otherwise. It should be clear that

The first instrument that will be used is the Leadership Questionnaire created by Melchar and Bosco (2010). There are 23 questions designed to determine the leadership styles of managers. All the survey questions are on a likert-scale and is to be given to level two managers within the organization. The second instrument is the Leadership

In these  assignment examine your current financial condition and life goals hence develop suitable retirement and estate plans. Financial condition and life goals:Retirement and estate plans Use various sources (including Prudential Retirement Planning) and course-related concepts to examine your current financial condition and life goals. Apply this knowledge to develop suitable retirement and estate plans. Your contingency

The fat edges in the graph G displayed in Figure 13.2 form a matching M. The vertices a, f, and y are exposed with respect to M, and the sequences (a, b, c, d, e, f) and (a, b, c, u, v,w, x, y) define augmenting paths P and, respectively. Interchanging the roles of edges and non-edges of M on the path yields the matching M_ of cardinality |M| + 1 exhibited