Báo cáo toán học: ": ALGEBRAIC MATCHING THEORY" pot

... 1. Introduction A k -matching in a graph G is a matching with exactly k edges and the number of k- matchings in G is denoted by by p(G, k). If n = |V (G)| we define the matchings polynomial µ(G, ... of vertices missed by a maximum matching in a graph G is the multiplicity of zero as a root of the matchings polynomial µ(G, x)of G, and hence many results in matching theory can be expresse...

Ngày tải lên: 07/08/2014, 06:20

Ngày tải lên: 05/08/2014, 15:20

Ngày tải lên: 05/08/2014, 15:20

... is a perfect matching of G then there is a unique perfect matching P  of G  such that x(P )=x  (P  ). Observe that sgn(D, P ∆Q)=sgn(D  ,P  ∆Q  ) for each pair of perfect matchings P, Q ... perfect matchings of G if and only if G is Pfaffian, and it means that we can decide efficiently whether a graph is Pfaffian once we can compute efficiently its number of perfect matchings. 2. Exact Mat...

Ngày tải lên: 07/08/2014, 06:20

... G. Definition 1.1 The generating function of the perfect matchings of G is the polyno- mial P(G, x) which equals the sum of x(P ) over all perfect matchings P of G. A subgraph G  =(V,E  ) of a graph ... 1998. Abstract This is a continuation of our paper “A Theory of Pfaffian Orientations I: Perfect Matchings and Permanents”. We present a new combinatorial way to compute the generating function...

Ngày tải lên: 07/08/2014, 06:20

... rainbow matching of size at least ⌈k/2⌉. We prove the slightly weaker statement that a rainbow matching of size at least ⌊k/2⌋ is guaranteed. We also give sufficient conditions for a rainbow matching ... Introduction Given a coloring of the edges of a graph, a rainbow matching is a matching whose edges have distinct colors. The study of rainbow matchings began with Ryser, who conjectu...

Ngày tải lên: 08/08/2014, 11:20

... relies upon Theorem 3.3. 4 Algebraic shifting Algebraic shifting transforms a simplicial complex into a shifted simplicial complex with the same f-vector, and also preserves many algebraic properties ... n 2 transcendentals, algebraically independent over k. Define f S := f i 1 ∧···∧f i k for S = {i 1 < ···<i k } (and set f ∅ =1). Let ∆(K):={S ⊆ [n]: ˜ f S ∈ span{ ˜ f R : R< L...

Ngày tải lên: 07/08/2014, 06:20

... {a n ,a n +2}. It is straightforward to confirm that  n i=1 TC n (i)=2 n . ✷ It remains to show that the hypotheses of Lemma 2 can be satisfied. Theorem 1 For al l n ≥ 1,thereisann-bit Gray code in which

Ngày tải lên: 07/08/2014, 06:20

... ∈N∪Pwhich lie on any directed path from some u 0 = u. (i) Let u 0 ∈P. By (2) and the induction hypothesis (5) applied to P (u 2 ), µP (u 0 ).∀u 1 ∈ F (u 0 )[∃u 2 ∈ F (u 1 )∀u 3 ∈ F (u 2 )N(u 3 )]. The ... part in brackets is (3) for u 1 , establishing (5). (ii) Let u 0 ∈N. By (3) and the induction hypothesis (6) applied to N(u 2 ), µN(u 0 ).∃u 1 ∈ F (u 0 )[∀u 2 ∈ F (u 1 )∃u 3 ∈ F (u 2 )P (u 3...

Ngày tải lên: 07/08/2014, 06:22

Ngày tải lên: 07/08/2014, 06:23