Báo cáo toán học: "The Rectilinear Crossing Number of K10 is 62" docx

... point is a vertex. Such a drawing is said to be a rectilinear drawing of G.Therectilinear crossing number of G,denoted ν(G), is the fewest number of edge crossings attainable over all rectilinear ... rectilinear drawing of K n is a triangle. Prove that the convex hull of any optimal rectilinear drawing of K n is a triangle. 2. Given a rectilinear drawing...

Ngày tải lên: 07/08/2014, 06:22

... ‘complexity’ of a graph decomposition D by the following parameters. The width of D is the maximum cardinality of a bag. The order of D is the number of bags. The degree of D is the maximum degree of the ... by cr(G), is the minimum number of crossings in a drawing 2 of G in the plane; see [13, 28, 29, 37, 48–50] for surveys. The crossing number is an impor...

Ngày tải lên: 07/08/2014, 15:22

... notion of the distinguishing number to group actions on sets and proves results about the distinguishing number of actions of the symmetric group S n . She shows that the distinguishing number of ... element. The distinguishing number of the action of G on X, denoted D G (X), is the smallest r admitting a distinguishing r-coloring of X with respect to the action of G...

Ngày tải lên: 07/08/2014, 13:21

... journal of combinatorics 17 (2010), #R59 17 α t (G) of G is the maximum order of a t-stable set in G. The main topic of this paper is to give a precise formula for the t-stability number of a dense ...  n α t,p (n) −2/ ln b −2 −ε − o(1) . 5.2 The lower bound This proof is the generalisation of a proof of the lower bound on the chromatic number of a dense random grap...

Ngày tải lên: 08/08/2014, 12:22

Ngày tải lên: 05/08/2014, 10:20

... = n is exercise 11.19 in [7]. The general case is similar. Proof. For the lower bound, we may assume that the union of all of the sets in C is [n]. The number of nodes in a directed forest is ... where L is the number of leaves and R is the number of roots. The maximal sets of C partition [n]sothereareat least n/k of them and the minimal sets are disjoint so...

Ngày tải lên: 07/08/2014, 13:21

... r-distinguishing (or just proper distinguishing) if it is a proper labeling (i.e., coloring) of the graph and no automorphism of the graph preserves all of the vertex labels. The distinguishing ... this is a proper (2∆−1)-distinguishing labeling of G and we are done. Otherwise, as before, y’s set of siblings is disjoint from its set of neighbors. In particular, the electroni...

Ngày tải lên: 07/08/2014, 13:21

... consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The t-pebbling number π t (G) of a graph G is the smallest m such that for every initial distribution of m ... (see [7]). All o f these are linear functions of t. Moreover, one can show that the t-pebbling number of any tree is linear in t by using the methods of [8]. It is shown in [6] t...

Ngày tải lên: 08/08/2014, 14:23

... consistent linear system Ax = b, (1) where A is an n×nM-matrix, b is an n dimension vector in rang(A). The study of the solution of the linear system (1) is very important in Leontief model of input-output ... such that Ax = b, then A is a nonsingular M-matrix. Proof. By Lemma 2.1, A is an M-matrix. Assume that A is singular. Then so is A t . Let A t = sI − B, s ∈ R and B...

Ngày tải lên: 06/08/2014, 05:20

... the polyominoes of area n. The perimeter P (c) of a polyomino c is the number of unit edges of the dual lattice ( +1/2) 2 which belong only to one of the unit squares of c. Notice that the perimeter is an ... M 13 \  M 13 In general, the set M n is much larger than  M n . It turns out that it is not the case for specific values of n. This is the content of the sec...

Ngày tải lên: 07/08/2014, 06:20