Báo cáo toán học: "Lower Bound for the Size of Maximal Nontraceable Graphs " pot

... not complete, then some internal vertex of Q has a neighbour in G − V (Q). the electronic journal of combinatorics 12 (2005), #R32 2 Lower Bound for the Size of Maximal Nontraceable Graphs ∗ Marietjie ... MNT graphs, which we shall use, in the next section, to prove that a MNT graph of order n ≥ 10 has at least 3n−2 2 edges. The first one concerns the lower boun...

Ngày tải lên: 07/08/2014, 13:21

... key ingredient of our proof. Given the informal discussion above, the proof of Theorem 1.1 looks very roughly as follows. At the first part of the proof we consider the graph G(n, ρ) for a relatively ... claim holds for d − 1. We prove the claim for d. Let S = (G 1 , G 2 , . . . , G d ) be the sequence of the labels of the nodes u i , i ∈ [d], along the path...

Ngày tải lên: 07/08/2014, 21:21

... consider d-fold grids over the class M of matchings. Theorem 8. For d ≥ 2, there exists G ∈ M d such that χ(G) ≥  d 6 log d  . Proof. As the proof is similar to the pro of of Theorem 7, we will g ... the function f(G; d) for the class G of k-colorable graphs and the class M of matchings. For the class S of graphs with at most one edge, we were not able to o...

Ngày tải lên: 07/08/2014, 21:21

... improving on the best previously known bounds k 3 (7) ≥ 153 and k 3 (8) ≥ 398. The proofs are founded on a recent technique of the author for dealing with systems of linear inequalities satisfied by the ... k ≈ n 2 . The reason is, that for larger values of k the “irregular” solutions of the linear inequalities no longer yield a negligible amount in the necessary es...

Ngày tải lên: 07/08/2014, 15:22

... sequence of connected graphs such that the values of the average genus of the graphs approach a fini te limit point. Then there exists an index N such that all but a finite num ber of graphs in the ... that the values of the average genus for 2-connected graphs have limit points. Note that the averag e genus for bar-amalgamation of a cactus and the g r ap...

Ngày tải lên: 08/08/2014, 12:23

... 5, µ 2 ∆ ≥  n t  4 p 2t 2 t 2  n t  2 p 2t 2 ϕ(1, 1) 10 the electronic journal of combinatorics 4 (1997), #R18 6 2. PROOFS Proof of the Theorem: We have already provided a proof of the second part of the theorem using nothing more ... holds for each r ≥ r 0 . 9 the electronic journal of combinatorics 4 (1997), #R18 3 Most of the work described in the previous t...

Ngày tải lên: 07/08/2014, 06:20

... problems on the square lattice L n . There the authors gave the following bounds for the asymp- totics of f(n), the number of forests of L n ,andα(n), the number of acyclic orienta- tions of L n :3.209912 ... lim n→∞ α(n) 1/n 2 . The proof of the second equality is very similar; it relies again on the formula T(G ∗ ; x, y)= T (G; y, x), and the fact that the...

Ngày tải lên: 07/08/2014, 07:21

... s  ,thepowerofy does not change in the passage from D  to D.Thepowerofw does not change either, since there are the same number of extra cells left of the last vertical move after adding the ... i,let(a, c) be the coordinates of the southeast corner of S i ,andlet(b, c) be the coordinates of the southwest corner of S i . First note that c = v 0 + v 1 + ···+ v i−1...

Ngày tải lên: 07/08/2014, 08:22

... A 0 . Moreover, we note that at the start of the i’th step, |{j < i | p j < p i }| columns to the left of the column p i in the original A 0 have already been removed. Therefore, at the i’th step, when ... by removing the first row and the j’th column (j = 1, 2, . . . , d+1) of A x . By the definition of T we see that the adjacency matrix of D is exactly T. B...

Ngày tải lên: 08/08/2014, 01:20

... either of these is the result of deleting the letters {3, 4, ,k− 4} from w,thenw must be either a word starting within the 0v 1,k−4 block and ending at the end of x k or a word starting at the ... occurrences of i from the left would be either the first and second or the second and third occurrences of i from the right. By the discussion following Lemma 3.3, the...

Ngày tải lên: 07/08/2014, 08:20