Báo cáo toán học: "Perfectly balanced partitions of smoothed graphs" doc

... a perfectly balanced partition almost surely. Smoothed Graphs. Smoothed analysis studies the behavior of objects after adding a small amount of randomness. This concept was introduced in the context of ... journal of combinatorics 16 (2009), #N14 6 • ∆(G) − δ(G) ≤ 2αn. • G has a degree matching of size at least 2αn. In the course of the proof of Lemma 5, we need the follow...

Ngày tải lên: 07/08/2014, 21:21

... ,k (9) Let B be the matrix of the coefficients of the (k +1)n + k equations (7)-(9). Lemma 11 Matrix B is of full row rank. Proof. We extend the first part of the proof of Lemma 6 to the case k>2. Each ... the matrix of the coefficients of the 4n + 1 equations (1)-(6). Lemma 6 Matrix A is of full row rank. Proof.Eacharc(j, n +1),j =0, ,n, of each of the two colours interfere...

Ngày tải lên: 07/08/2014, 08:22

Ngày tải lên: 05/08/2014, 09:46

Ngày tải lên: 05/08/2014, 09:46

... cycles. Let ¯ d be the average degree of vertices of G.Then α(G) ≥  n ¯ d/2. Proof. Since G contains no 3 or 5 cycles, we have α(G) ≥ d 1 (v) (consider the neighbors of v)andα(G) ≥ 1+d 2 (v) (consider ... 2 − ( k−1 k ) n k−1 k d 1 k . Proof. Immediate from Theorem 3. This improves Denley’s Theorem 4. References 1. Denley, T., The Independence number of graphs with large odd girth, Th...

Ngày tải lên: 07/08/2014, 06:20

... (V C ,C)ofG(V,E). The set C of all cycles forms a subspace of (E, ⊕, ·) which is called the cycle space of G.AbasisBof the cycle space C is called a cycle basis of G(V, E) [2]. The dimension of the ... minimal cycle basis of G. The remainder G ∗ of G after all cycles C β , β ∈ K are removed by the above procedure is composed of Y ∗ and those edges of H that are not contained...

Ngày tải lên: 07/08/2014, 06:22

... edge-connectivity of a graph G is the size of a smallest odd edge cut of G.The odd girth of G is the length of a shortest odd cycle of G. Zhang [19] proposed a strengthing of Jaeger’s conjecture, ... +1,k)-colouring of G  can be extended to a (2k +1,k)- colouring of G. Proof. Assume that x is a vertex of G with d l (x) ≥ 2k(d(x)−1). If d(x)=2,thenx is the internal vertex...

Ngày tải lên: 07/08/2014, 06:22

... as the product of all parts of all partitions of n. It also equals b n , which is defined as the product of all factorials of all multiplicities of parts in partitions of n. Proofs of this may be ... number of partitions of n. The number p ∗ (n) of regular partitions of n equals the number of class regular partitions of n and then p  (n)=p(n) − p ∗ (n) is...

Ngày tải lên: 07/08/2014, 07:21

... coloring of H l . • If c(w  )=c(w + ) = c(w ⊕ ), then c can be extended to exactly l different proper colorings of H l . The number of vertices of H l does not exceed 3+2log 2 l. Proof: The proof ... number of variables and clauses of the formula Φ. In particular, the number of vertices of H is at most 3 · max A 2 +2·|A 2 \ A 1 |·n and the number of its edges is cubic in...

Ngày tải lên: 07/08/2014, 08:20

... convenience, we will often speak of “the” adjacency matrix of Γ when it is clear that a specific choice of an ordering of the vertices of Γ is inconsequential. 2.2 Linear Complexity of a Matrix Let ... linear complexity of a matrix, and we introduce the notion of the linear complexity of a graph. We also see how we may relate the linear complexity of a graph to that of on...

Ngày tải lên: 07/08/2014, 13:21