... can quickly complete the proof of the main theorem using the result about
the special graph Λ proven in the next section.
Proof of Theorem 4. If H = S
k
,thenanyk-coloring of S
n
, n ≥ k induces ... star are isomorphic. Then, by Claim 0, ∆(G) ≤ k −1.
Consider a set S of vertices of G inducing H and let S
⊆ S correspond to the set of leaves
in H.Letl be the larges...
... provide the exact value of the Ramsey numbers R(P
n
1
, P
n
2
, . . . , P
n
t
, C
k
)
for certain values of n
i
and k and then we determine the exact values of some three-color
Ramsey numbers ... 5, R(P
4
, C
4
, P
m
) = m + 2.
Proof. Using Theorem 3.8, we have R(P
4
, C
4
, P
m
) ≤ m + 2. On the other hand, the
3-colored graph K
m+1
in the proof of Theorem 3.8, implie...
... ⋂
bÎΓ
X
b
. The proof of Theorem 3.1 is therefore complete.
As a consequence of this theorem, we obtain the following common fixed point
result.
Theorem 3.2. Let X be a comple te lattice. Then, an ... family (F
b
)
bÎΓ
is
decreasing . Theorem 3.1 implies that ⋂
bÎΓ
F
b
is nonempty and is a complete su blattice
of X. The proof of Theorem 3.2 is therefore complete.
The commuta...
... initially
(0). Then, at each of the n following stages of the BFS, the vertex x at the head of the
queue is removed from the queue, and all “new” neighbors of x areaddedattheendof
the queue, in increasing ... distribution of ˆy
(n)
given that V = n is the distribution
of the sequence z
(n)
under the hypothesis of equiprobability of confined sequences, that...
... cycle, such that there are f
i
i-gons in the exterior
of the Hamiltonian cycle and f
′
i
i-gons in the interior of the Hamiltonian cycle. Then
i
(i − 2)(f
i
− f
′
i
) = 0. (1)
The theorem can be ... and planar. Both have one face of degree 4,
and four faces o f degree 7, such that the face of degree 4 is adjacent to all faces of degree
7 and the degrees of...
... partition P of [1,n], we define the depth, denoted d(P ), to be the
largest of the set minimums; and for a symmetric sum-free partition we define
the e-depth, denoted D(P), to be the maximum of d(P
) ... are no symmetric sum-free partitions of 155 or 158 into
5sets.
Note 155 and 158 are the special cases in the definition of symmetric, and that
it is not obvious t...
... immune cells (B and T lymphocytes and macrophages)
and/ or the produc tion of antibodies directed against
myelin components. These events lead to the damage and
degeneration of the myelin sheath. ... sites, where the ECM binds to a family of receptor
proteins within the cell membrane. The most significant of
these are the integrins. Binding of ECM ligands to...
... study on the problem of computing the difference between
the metric dimension and the determining number of graphs. We provide new proofs
and results on the determining number of trees and Cartesian ... 6).
P
2
P
3
P
4
P
5
P
3
P
2
Figure 6: Metric basis of P
t
P
m
.
The connection between the determining number and the metric dimension, an...