Báo cáo toán học: "Minimal percolating sets in bootstrap percolation" pps
Báo cáo toán học: "Minimal percolating sets in bootstrap percolation" pps
... that in Lemma 4, below, we need to use three extra
columns to ‘connect’ two corner-avoiding minimal percolating sets.
The next lemma explains our interest in corner-avoiding minimal percolating sets.
Lemma ... a minimal percolating set
(MinPS) if A percolates in r-neighbour bootstrap percolation, but no proper subset of A
percolates. Clearly a percolating set of minimal si...
Báo cáo toán học: " On Score Sets in Tournaments" pptx
... arithmetical
analysis which appeared in Chinese [6] in 1988 and in English [7] in 1989.
In the following result, we prove that any set of p non-negative integers
s
1
,s
2
, ,s
p
with s
1
<s
2
< ... Yao, Reid’s conjecture on score sets in tournaments, Kexue Tongbao 33
(1988) 481–484 (in Chinese).
7. T. X. Yao, On Reid’s conjecture of score sets for tournaments, Chinese S...
Báo cáo toán học: "Perfect dominating sets in the Cartesian products of prime cycles" docx
... and
contains at least one perfect dominating set, a set S ⊆ V (G) is a minimum dominating
set if and only if it is a perfect dominating set. In [2] Chartrand et al. studied the size
of defining sets ... a minimum dominating set of C
n
2n+1
, the Cartesian
product of n copies of the cycle of size 2n + 1, where 2n + 1 is a prime.
Keywords: Perfect Lee codes; dominating sets; defining sets....
Báo cáo toán học: "Maximum Independent Sets in Certain Powers of Odd Cycles" doc
... independent sets in C
2
4ℓ+1
was the key to obtaining the upper bound on α(C
3
8k+ 5
) in
[2]. In other related work, the authors exploited structural properties of near maximum
independent sets ... main result is that the collection of independent sets defined in Lemma 1 is, up
to isomorphism, the complete list of maximum independent sets in C
d
k2
d
+1
.
the electronic jour...
Báo cáo toán học: "Minimal Cycle Bases of Outerplanar Graphs" ppsx
... is minimal,
since every cycle has length 3. But it is non-fundamental, since every edge is covered
at least two times.
The concept of fundamental has originally been introduced by Kirchhoff in ... (B)=
C∈B
|C|.Aminimal cycle basis
is a cycle basis with minimal length. Let c(B) be the length of the longest cycle in
the cycle basis B. Chickering [3] showed that if (B) is minimal then c(B...
Báo cáo toán học: "Longest Induced Cycles in Circulant Graphs" pps
... maximum lengh of a cycle
in unit circulant graphs on a Dedekind domain quotiented by an ideal with r distinct
maximal factors. Dedekind domains are exactly those integral domains in which every
ideal ... of combinatorics 12 (2005), #R52 10
A Dedekind ring is simply a quotient of a Dedekind domain.
If R is a Dedekind domain, and m
i
is a maximal ideal of R,thenR/m
i
is a field and
thus cont...
Báo cáo toán học: " Degree constrained orientations in countable graphs" pps
... not, direct f
1
in the other way, and define u
1
accordingly.
Lemma 7 ensures that G
1
= G
0
− f
1
has no u
1
-faulty sets. Continuing in this way, we
obtain the desired orientation. Indeed, suppose ... vertex v of in nite degree we can only demand all
or nothing. Setting u(v) to a finite value in Theorem 2 is the same as requiring in nitely
many outgoing edges at v, whereas putting u(...
Báo cáo toán học: "An extremal theorem in the hypercube" pps
... edge formed by replacing b
g(i
ℓ
)
in f(u) with a flip-bit is in G. But this edge is just the
edge between f(u) and f(v). We therefore have an embedding of H in G, completing the
proof. ✷
To complete ... chromatic number of H. In
particular, it is known [16] that ex(K
n
, H) = o(e(K
n
)) if and only if H is bipartite. In
this note, we will be similarly interested in determining sub...
Báo cáo toán học: "Embedding a Forest in a Graph" ppsx
... embedded in any graph with mini-
mum vertex degree d. Indeed, a linear algorithm to find such an embedding would sequentially
embed the vertices of the tree according to a depth first search ordering ... sizes are a
1
, . . . , a
p
can be
embedded in any graph containing at least
p
i=1
(a
i
+ 1) vertices and having minimum
degree at least
p
i=1
a
i
.
1 Introduction
It is a folklore fact...