Báo cáo toán học: "New infinite families of almost-planar crossing-critical graphs" ppsx

... construct new infinite families of 3-designs. 1 Introduction A t − (v, k, λ) design is a pair (X, B) where X is a v-element set of points and B is a collection of k-element subsets of X called blocks, ... Jangheon Oh and Hoseog Yu, New infinite families of 3-designs from algebraic curves over F q , European J. Combin. 28 (2007), no. 4, 1262–1269. [10] K. Ranto, Infinite fami...

Ngày tải lên: 07/08/2014, 15:22

... communication and preprint of [2]) that typi- cal constructions of infinite families of simple 3-connected k -crossing-critical graphs create bounded numbers (wrt. k) of vertices of degrees other than ... in infinite families of k -crossing-critical graphs? We positively answer one half of his question in Theorem 3.1 and Proposition 2.1; • namely we construct, for all k...

Ngày tải lên: 07/08/2014, 21:20

... design in the infinite family has the collection of five blocks mentioned above then so do all other members of the infinite family; 3) Give a design having the required collection of five blocks. Theorem ... matrix ofa2-(v, k, λ)designisab × v matrix A = a ij ,in which a ij =1ifblocki contains point j and a ij =0otherwise. 3 Results The strategy for establishing the existence of an...

Ngày tải lên: 07/08/2014, 06:20

... partially ordered set of finite permutations ordered by deletion of entries contains an infinite antichain. In other words, there exists an infinite collection of permutations no one of which contains ... than the poset of graphs ordered by the operations of edge contraction and vertex deletion. (That this poset of graphs does not contain an infinite antichain is a famous theo...

Ngày tải lên: 07/08/2014, 06:20

... structural properties and that the natural way to describe the structure of these graphs is in terms of graph homomorphisms. One infinite sequence of cores of dense maximal triangle-free graphs was known. All graphs ... 4-colourable, in contrast to a conjecture of Jin [8], saying that there are graphs of arbitrarily large chromatic number with this property. Another infinite seq...

Ngày tải lên: 07/08/2014, 06:22

... derivative of Z X (u) is a generating function for the number of circuits. See for instance [7]. Knowing the number of circuits of each length is sufficient to conclude the number of primitive circuits of ... demonstrate add itional families of pairs of graphs which share the same zeta function. 1. Introduction In 2009 , Cooper described an infinite family of non-isomorphic...

Ngày tải lên: 08/08/2014, 12:22

... 1, again contradicting the choice of G. This completes the proof of Claim 3.1. ✷ In view of Claim 3.1 and Lemma 3.1, B must be K 4 -minor-free. By the proof of Claim 2.1, B  ∼ = K 2 , so that ... that every edge of G is given a list of ∆ colours. Then the edges of G − (B − z 0 ) can be properly coloured from these lists. Since each edge of B still has a residual list of at...

Ngày tải lên: 07/08/2014, 13:21

... decomposition of K k+1 ∗ K m whenever k is odd and m is even . Proof As in the proof of L emma 3.12, we need only give a k-cycle decomposition of 2K k+1 which satisfies the conditions of Corollary ... vertices. In fact, since the sum-weight of each cycle in D is a multiple of λ, each of these subgraphs necessarily consists of λ pairwise vertex-disjoint k-cycles, each of which...

Ngày tải lên: 08/08/2014, 01:20

... prefix of x of length 5. Proof. Firstnotethatthewordyis necessarily overlap-free by assertion 1 of Lemma 2. The existence of the decomposition for finite words is proved in [10], in the proof of Theorem ... this factorization of x,thewordu(resp. v) depends only on the prefix (resp. suffix) of x of length 7. This is recalled in the following table of all possible prefixes (resp. su...

Ngày tải lên: 07/08/2014, 06:22

... a tree. Proof: It suffices to prove that every edge of T (G) is an isthmus. Corollary 9 shows that if [x, y] is any hinge of G, then the cleavage units of G are precisely the cleavage units of the ... of K,oneofH and J is 2-connected. It is straightforward to show the corresponding one of H  and J  is also 2-connected, so that (H  ,J  ) is a hinge-separation of G. But then eve...

Ngày tải lên: 07/08/2014, 08:20