Báo cáo toán học: "Some gregarious cycle decompositions of complete equipartite graphs" ppsx

... pairs of cycles in such a way that the cycles in each pair share two adjacent edges . Then there exists a gregarious k -cycle decomposition of G ∗ K 2 . Proof Let D be a k -cycle decomposition of ... to “nongregarious” cycle decompositions, few results are known. Of course 3 -cycle decom- positions of complete equipartite graphs are necessarily gregarious and so by Han...

Ngày tải lên: 08/08/2014, 01:20

... 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 ... vertices of degree 2, by Theorem 1.3. This contradiction completes the proof of Theorem 1.2. ✷ the electronic journal of combinatorics 13 (2006), #R98 6 Edge and total choosability of...

Ngày tải lên: 07/08/2014, 13:21

... write uniquely each element of H i as the sum of an element of K 1 i together with an element of K 2 i , with K 1 i a suitable subgroup of < d > and K 2 i a suitable subgroup of < 2 m >, i.e., ... union of cycles. Namely: if the o r der of h in G is t, then the cycles are given by (x, hx, h 2 x, . . . , h t−1 x), as x varies in the set of distinct representatives...

Ngày tải lên: 08/08/2014, 12:22

... R (n) j+1 of [0, 1] n is partitioned, up to a set of measure zero, by the images of the interiors of the simplices ∆ σ for those signed permutations satisfying des(σ) = j. Proof. The former part of ... R (n−1) i of ∆ n−1 is a projection of the section S (n) i of ∆ n . Proof. First, we describe the vertices of R (n−1) i when i satisfies 2 ≤ i ≤ n −1. These vertices fall into t...

Ngày tải lên: 07/08/2014, 06:22

... b)-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms an H-subgraph. We assign a weight of b to each H-subgraph in the decompo- sition and a weight of ... part. We assign a weight of b to each H-subgraph in the decomposition and a weight of 1 to single edges. The total weig ht of the deco mposition is the sum of the we...

Ngày tải lên: 08/08/2014, 14:23

... second statement of Theorem 1. ProofofStatement(ii) of Theorem 1. By the same argument in the proof of Stament (i) of Theorem 1, we can assume without loss of generality that A is complete. Assume ... ideal of M  and let (x 1 , ,x d ) be a system of parameters of M  such that q  =(x 1 , ,x d )A. Then the lemma is proved if we can show the existence of a system of para...

Ngày tải lên: 06/08/2014, 05:20

... types: 1. a 12 -cycle, 2. the disjoint union of a 9 -cycle and a triangle, 3. the disjoint union of two 6-cycles, 4. the disjoint union of a 6 -cycle and two triangles, 5. the disjoint union of four triangles. One ... each pair of non- adjacent neighbours has 4 common neighbours. In [3] an incomplete enumeration of strongly regular (40, 12, 2, 4) graphs established the existenc...

Ngày tải lên: 07/08/2014, 06:20

... the next operation of Type i. We proceed with an examination of each of the two phases before giving a formal definition of the distinction between the phases. For a clutch of operations in each ... least to within an additive constant, of the main result of [10]. They also showed that the size of a connected dominating set of an n-vertex cubic graph of girth at least 5 is...

Ngày tải lên: 07/08/2014, 06:23

... most of the graphs generated have the n-e.c. property in Section 4, thereby completing the proof of Theorem 1.1. The proof of the n-e.c. property uses bounds on the expected number of pairs of ... characters as in the proof of Theorem 10, Section XIII.2, of Bollob´as [2]. If U and W are disjoint sets of vertices of the Paley tournament → P q ,then we denote by v(U, W ) the n...

Ngày tải lên: 07/08/2014, 06:23

... replacements α α α α γ γ γ γ γ β β β β Figure 1: The colouring of H used in the second case of the proof of Lemma 1 when n ≡ 1 (mod 3) 3.2 Proof of Lemma 1 Fix d ≥ 5, n ≥ 6, and let G be a (d − 2)-regular ... deferred to Section 4. the electronic journal of combinatorics 13 (2006), #R77 2 Let us see now how these two lemmas yield the proof of Theorem 2. Proof of Theorem 2. Note...

Ngày tải lên: 07/08/2014, 13:21