Báo cáo toán học: "Counting Connected Set Partitions of Graphs" pot

... of a connected set partition form a cut set. An edge cut of G corresponds to a cut set defined by a connected two-block partition. In order to keep track of the number of edges forming a cut set, ... is denoted by |π| and the set of all set partitions of X by Π(X). The number of set partitio ns of a n n-element set with exactly k blocks is called the Stirlin...

Ngày tải lên: 08/08/2014, 12:23

... tudy of minimally intersecting set partitions, we investi- gate minimally intersecting set partitions of type B. Our main result is a formula for the number of m inimally intersecting r-tuples of ... structure of the lattice of type B n parti- tions of the set {±1, ±2, . . . , ±n}. The lattice of type B n set partitions has been studied by Reiner [8]. It can be re...

Ngày tải lên: 08/08/2014, 11:20

... -assignment of G is a mapping L which assigns to each vertex v of G a set L(v) of f(v) positive integers as permissible colours. A g-colouring of G is a mapping S which assigns to each vertex v of G a set ... only defined for a subset U of V (G), then it is assumed that f(x) = 0 for x /∈ U. • For a subset X of V (G), f (X) =  v∈X f(v). • For a subset X of V , G[X] is the s...

Ngày tải lên: 08/08/2014, 01:20

... numb er of singleton blocks equal to {1} in partitions of [n] is equal to the number of partitions of [n − 1] we see that the generating function for the number of singleton blocks in all partitions ... differentiation of exponential generating functions only means a shift of coeffi- cients, we find that the total of sumrec, summed over all set partitions of n into k s...

Ngày tải lên: 08/08/2014, 12:22

... least to within an additive constant, of the main result of [10]. They also showed that the size of a connected dominating set of an n-vertex cubic graph of girth at least 5 is at most 2n/3+O(1). ... is referred to [8]. The algorithms of [4, 7], that find a small connected dominating set of cubic graphs, guarantee that the size of the connected dominating set return...

Ngày tải lên: 07/08/2014, 06:23

... σ of the 1st through (b − (q − 1))th copies of k (the positions of the last q − 1 copies of k do not affect the number of occurrences of the pattern), a nd we let A = a b−(q− 1) . The number of ... one of the first p letters is k. By the principle of inclusion-exclusion, the number of such σ is p  m=1 N m (−1) m+1 , where N m is the sum, over all m-subsets of the first p p...

Ngày tải lên: 08/08/2014, 14:23

... concrete example, let S bethesetofallwordsoflength7overΣ={a, b}. Each word of S is 2-avoidable, but any binary word of length 7 or more encounters an element of S. An image of xxx or xyxyx under a ... prefix of m n has length at most 11, and is a prefix of m 1 . Inspection shows that no prefix of m 1 of length 11 or less is an overlap.✷ Theorem 2.10 The set {m n } n∈IN is an an...

Ngày tải lên: 07/08/2014, 06:20

... electronic journal of combina torics 12 (2005), #N23 5 Now we are ready to show the main results. We start with the proof of Theorem 2 and then derive the proof of Theorem 1 from it. Proof of Theorem ... 2. Proof. For any edge e of G and any vertex u ∈ V (G) −V (e), let the neighbor vertices of u be x 1 ,x 2 , ,x 2r . We construct a new graph G ∗ by using two copies of G − u and...

Ngày tải lên: 07/08/2014, 08:22

... more than k vertices and remains connected after deletion of any subset of k−1 vertices. Furthermore, we use two basic facts of graph-theory: the faces of a two -connected planar embedding are bounded ... component G 2 of G O \ V 2 so that G 2 ∪ V 2 is a two -connected subgraph of G O \ G 1 . 2. V 1 consists of two vertices and there is a subset V 2 of two vertices and a c...

Ngày tải lên: 07/08/2014, 13:21

... Since the set of bags of D 1 that belong to a specific bag of D induces a connected (cycle) subgraph of D 1 , and D(v) is a connected subgraph of D for each vertex v of G, D 1 (v) is a connected ... proof of Theorem 2 in this paper. Let G and D be graphs, such that each vertex of D is a set of vertices of G (called a bag). Note that we allow distinct vertices of...

Ngày tải lên: 07/08/2014, 15:22