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...

Ngày tải lên: 07/08/2014, 21:21

... 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...

Ngày tải lên: 06/08/2014, 05:20

... 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....

Ngày tải lên: 07/08/2014, 13:21

... 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...

Ngày tải lên: 07/08/2014, 21:21

Ngày tải lên: 05/08/2014, 15:20

... 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...

Ngày tải lên: 07/08/2014, 06:22

... 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...

Ngày tải lên: 07/08/2014, 13:21

... 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(...

Ngày tải lên: 07/08/2014, 21:20

... 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...

Ngày tải lên: 08/08/2014, 12:22

... 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...

Ngày tải lên: 08/08/2014, 14:23