... 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...
... 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: ... underlying graph of G
σ
, and we denote by Sp(G) the adjacency spectrum
of G. S kew-adjacency matrix S(G
σ
) of G
σ
is intro duced, and its spectrum Sp
S
(G
σ
)
is ca...
... 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...
... 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...
... 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...
... 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...
... 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...
... 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...
... there exists π
∈ X
which is a deflation of π by permutations in Y . The proof of Lemma 3.1 then tells us
that π
Y
≤ π
, completing the proof.
Any expression of the form π = π
Y
[α
1
, . . . , α
k
] ... avoid
the permutation 321 and so the antichain in the proof of Theorem 6.1 lies in the
basis of Av(25134) Y in both cases.
(ii) All of the classes of the form Y = Av(α, β) wh...
... 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...