Báo cáo toán học: "Sharply transitive 1-factorizations of complete multipartite graphs" pot

... several properties of a maximal IC-coloring f of K m,n . Proposition 3.3. If f is a maximal IC-coloring of G = (A, B) = K m,n , then all the colorings of vertices of G are distinct. Proof. Suppose, ... proof is complete. the electronic journal of combinatorics 15 (2008), #R43 2 n ≥ 3, we have f 4 = 10 ≤ 3 · 2 4−2 − 2 4−(n+2) − 1. This is a contradiction to that f is a maximal IC...

Ngày tải lên: 07/08/2014, 15:23

... by the induction hypothesis, and this L-edge-coloring can be extended to an L-edge-coloring of G, because every edge of C is incident with at most k − 2edgesofG  . If (d) of Lemma 2.1 holds, ... edge e ∈ E(G). By Lemma 2.1 one of (a)–(d) of that lemma holds. If (a) holds, then the theorem follows by induction applied to the graph G − v,wherevis a vertex of G of degree at most o...

Ngày tải lên: 07/08/2014, 06:20

... multiplicity of the eigenvalue λ of X n , then λα(λ) is a multiple of φ(n). 3. ENERGY OF UNITARY CAYLEY GRAPHS We first give a direct proof of the result of Balakrishnan [1] when n is a power of a prime. Theorem ... length m which consists of all nonzero eigenvalues of X n such that the number of appearances of an eigenvalue is its multiplicity. 2. The multiplicity of z...

Ngày tải lên: 07/08/2014, 21:21

... proof of this last statement is quite simple, see Lemma 2.1 of [7] or Proposition 2.2 of [10] for instance. In this paper we focus our attention on complete multipartite graphs. We denote a complete ... this manner we complete Σ ′ to a starter for (G, Ω).  3 Sharply transitive 1-factorization s of bi partite complete graphs We know that the existence of a G−reg ular 1−f...

Ngày tải lên: 08/08/2014, 12:22

Ngày tải lên: 05/08/2014, 10:20

... of arcs of H 1 and B a subset of arcs of H 2 . We denote by CR(A × B)the number of crossings between arcs of A and B, mod 2. We also denote by cr(i, j)the number of crossings of arcs of H i ∩ ... sign of the permutation i 1 j 1 i n j n of 12 (2n). Each nonzero term of the expansion of the Pfaffian of A(D) equals x(P ) or −x(P ) where P is a perfect matching of G.I...

Ngày tải lên: 07/08/2014, 06:20

... whose bias corresponds to the ratio of the cardinalities of two distributive lattices, namely, the lattice of order ideals of P containing x and the lattice of order ideals of P not containing x. This ... proof of a formula for the number of Young Tableaux of a given shape, Adv. in Math. 31 (1979), 104–109. [GNW2] C. Greene, A. Nijenhuis and H. Wilf, Another probabilistic me...

Ngày tải lên: 07/08/2014, 06:22

... illuminating combinatorial proof of (3.1 ), which is an adapta- tion of Franklin’s proof of the pentagonal number theorem to this case (cf. [1] or [4]). Let u beawordoflengthn − 1,u= a n−1 . Then ... simple proof of (3.2 ). Morse code sequences are finite sequences of dots (•) and dashes (−). We assume that a dot has length 1 and a dash has length 2. The number of all such sequenc...

Ngày tải lên: 07/08/2014, 07:21

... equivalent to diversity To complete the proof of our upper bound, we must complete the proof of Theorem 3.3, whose statement we recall here. Theorem 3.3. Let X ⊂ (Z/pZ) 2 be a complete cap. The following ... consequence of the proof of Proposition 4.2. We now begin the proof of Theorem 3.3. In the following proposition, we prove the implication (1) ⇒ (2). Proposition 4.2. Let...

Ngày tải lên: 07/08/2014, 13:21

... copies of matrices. This leads to two types of evaluation formulas for the permanent of copies of matrices A . One in terms of certain orientations of A , the other in terms of colorings of A . ... vertices of the line graph ( interchange graph) LG of a graph G are the edges LG of G , V (LG):=E(G) . The number of edges between two vertices v,v  ∈ V (LG)in the line gra...

Ngày tải lên: 07/08/2014, 13:21