Báo cáo toán học: "Skew Spectra of Oriented Graphs" potx

... University Press, 1987. the electronic journal of combinatorics 16 (2009), #N32 6 Skew Spectra of Oriented Graphs Bryan Shader Department of Mathematics University of Wyoming, Laramie, WY 82071-3036, USA email: ... σ. Proof. (Necessity) See the proof of Theorem 3.3 in [1]. (Sufficiency) Suppose Sp S (G σ ) = iSp(G) for any orientation σ. By Theorem 2.2, G is a bipartite graph. And...

Ngày tải lên: 07/08/2014, 21:21

... member of the second coordinate of the label of y, delete the edge x j y.If the digit of z is 3 and y is not a member of the second coordinate of the label of z, delete the electronic journal of ... Subclaims, the proof of our lemma is complete. 5 Proof of the Main Theorem In this section, we present the proof of the main theorem. Let C be a topologically closed class o...

Ngày tải lên: 07/08/2014, 06:23

... δ) denote the set of neighbours and the degree of v in G, the set of vertices, the set of edges and the minimum degree of G, respectively. By G[D] we mean an induced subgraph of G with the vertex ... V 2 , . . . , V k } of disjoint subsets of a set V is called a partition of V if the union of all elements of V is V and V i = ∅ for every i. We shall denote as P k a path...

Ngày tải lên: 07/08/2014, 21:20

... one of size 2n − s. The rows and columns of an H(s, 2n) form orthogonal 1-factorizations of the underlying graph. As with Room frames, the existence of a pair of orthogonal 1-factorizations of ... fr om S, 2. each symbo l of S occurs once in each row and column of H, and 3. every unordered pair of symbols occurs in at most one cell of H. We note that a Room square of sid...

Ngày tải lên: 08/08/2014, 12:22

... Subsquares of N 2 squares of order 9. In particular we note that there are 1589 main classes of N ∞ squares of order 9, yet only 37 of these contain pan-Hamiltonian squares. In terms of reduced ... have a generalisation of Lemma 4. §6. Small orders For n ∈{2, 3, 5} thecatalogueofLatinsquaresin[5]showsthatthereisasinglemain class of N ∞ square of order n,namelythatofC n . Acc...

Ngày tải lên: 07/08/2014, 06:20

... sequence of types of the positions wi for i ≥ 1 (we think of this as the right phase diagram of w.) Proposition 4 The right phase diagram of w consists either of : • a string (possibly empty) of N ... full phase diagram of w which consists of the types of all the strings of the form awb where a, b > 0. We write this diagram as a quarter infinite array, with the value...

Ngày tải lên: 07/08/2014, 06:23

... ordering of the vertices to an ordering of the edges lexicographically. A broken circuit of H ⊆ G is a set of edges B ⊆ H such that there is some edge e ∈ G, smaller than every edge of B, such ... consists of all edges of G with both ends in S. We will use the symbols π and σ to denote set partitions. The notation π  S means π is a set partition of the set S. The length (numb...

Ngày tải lên: 07/08/2014, 08:22

... one-factorization of type 4 + 4 + 4 is only possible for three graphs of order 12: the disjoint union of three copies of C 4 , the disjoint union of three copies of K 4 ,and the union of K 4 and the ... properties of the classified one-factorizations are also tabulated. 1 Introduction An r-factor of a graph G is an r-regular spanning subgraph of G.Anr-factorization of G i...

Ngày tải lên: 07/08/2014, 08:22

... each face boundary of K 4 and changing the parity of the lengths of the paths of J corresponding to the chosen edges. The set of chosen edges of K 4 is a non-empty cocycle of K 4 since the face ... G. If H is a subgraph of the embedded graph, then H ∗ is the subgraph of G ∗ consisting of the dual edges of H. A bond of G is the minimal nonempty edge cut. To each bond o...

Ngày tải lên: 07/08/2014, 13:21

... and at least one of them contains an end of an edge of A  i ∩ B  since at least one of G[X i ∩ Y ] and G[X i ∩ Y ] contains an end of an edge of A i ∩ B. the electronic journal of combinatorics ... G ∗ is necessarily connected. Each bond of G is a circuit of G ∗ , and each circuit of G is a bond of G ∗ . If a theorem holds for a given set T of edges of G, then the...

Ngày tải lên: 08/08/2014, 11:20